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/*
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* DO NOT ALTER OR REMOVE COPYRIGHT NOTICES OR THIS FILE HEADER.
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package java.lang;
import java.math.BigDecimal;
import java.util.Random;
import jdk.internal.math.FloatConsts;
import jdk.internal.math.DoubleConsts;
import jdk.internal.vm.annotation.IntrinsicCandidate;
import static java.lang.Double.*;
/**
* The class {@code Math} contains methods for performing basic
* numeric operations such as the elementary exponential, logarithm,
* square root, and trigonometric functions.
*
* <p>Unlike some of the numeric methods of class
* {@link java.lang.StrictMath StrictMath}, all implementations of the equivalent
* functions of class {@code Math} are not defined to return the
* bit-for-bit same results. This relaxation permits
* better-performing implementations where strict reproducibility is
* not required.
*
* <p>By default many of the {@code Math} methods simply call
* the equivalent method in {@code StrictMath} for their
* implementation. Code generators are encouraged to use
* platform-specific native libraries or microprocessor instructions,
* where available, to provide higher-performance implementations of
* {@code Math} methods. Such higher-performance
* implementations still must conform to the specification for
* {@code Math}.
*
* <p>The quality of implementation specifications concern two
* properties, accuracy of the returned result and monotonicity of the
* method. Accuracy of the floating-point {@code Math} methods is
* measured in terms of <dfn>{@index ulp}s</dfn>, {@index "units in
* the last place"}. For a given floating-point format, an
* {@linkplain #ulp(double) ulp} of a specific real number value is
* the distance between the two floating-point values bracketing that
* numerical value. When discussing the accuracy of a method as a
* whole rather than at a specific argument, the number of ulps cited
* is for the worst-case error at any argument. If a method always
* has an error less than 0.5 ulps, the method always returns the
* floating-point number nearest the exact result; such a method is
* <dfn>correctly rounded</dfn>. A {@index "correctly rounded"}
* method is generally the best a floating-point approximation can be;
* however, it is impractical for many floating-point methods to be
* correctly rounded. Instead, for the {@code Math} class, a larger
* error bound of 1 or 2 ulps is allowed for certain methods.
* Informally, with a 1 ulp error bound, when the exact result is a
* representable number, the exact result should be returned as the
* computed result; otherwise, either of the two floating-point values
* which bracket the exact result may be returned. For exact results
* large in magnitude, one of the endpoints of the bracket may be
* infinite. Besides accuracy at individual arguments, maintaining
* proper relations between the method at different arguments is also
* important. Therefore, most methods with more than 0.5 ulp errors
* are required to be <dfn>{@index "semi-monotonic"}</dfn>: whenever
* the mathematical function is non-decreasing, so is the
* floating-point approximation, likewise, whenever the mathematical
* function is non-increasing, so is the floating-point approximation.
* Not all approximations that have 1 ulp accuracy will automatically
* meet the monotonicity requirements.
*
* <p>
* The platform uses signed two's complement integer arithmetic with
* {@code int} and {@code long} primitive types. The developer should
* choose the primitive type to ensure that arithmetic operations
* consistently produce correct results, which in some cases means the
* operations will not overflow the range of values of the
* computation. The best practice is to choose the primitive type and
* algorithm to avoid overflow. In cases where the size is {@code int}
* or {@code long} and overflow errors need to be detected, the
* methods whose names end with {@code Exact} throw an {@code
* ArithmeticException} when the results overflow.
*
* <h2><a id=Ieee754RecommendedOps>IEEE 754 Recommended
* Operations</a></h2>
*
* The 2019 revision of the IEEE 754 floating-point standard includes
* a section of recommended operations and the semantics of those
* operations if they are included in a programming environment. The
* recommended operations present in this class include {@link sin
* sin}, {@link cos cos}, {@link tan tan}, {@link asin asin}, {@link
* acos acos}, {@link atan atan}, {@link exp exp}, {@link expm1
* expm1}, {@link log log}, {@link log10 log10}, {@link log1p log1p},
* {@link sinh sinh}, {@link cosh cosh}, {@link tanh tanh}, {@link
* hypot hypot}, and {@link pow pow}. (The {@link sqrt sqrt}
* operation is a required part of IEEE 754 from a different section
* of the standard.) The special case behavior of the recommended
* operations generally follows the guidance of the IEEE 754
* standard. However, the {@code pow} method defines different
* behavior for some arguments, as noted in its {@linkplain pow
* specification}. The IEEE 754 standard defines its operations to be
* correctly rounded, which is a more stringent quality of
* implementation condition than required for most of the methods in
* question that are also included in this class.
*
* @spec https://standards.ieee.org/ieee/754/6210/
* IEEE Standard for Floating-Point Arithmetic
*
* @since 1.0
*/
public final class Math {
/**
* Don't let anyone instantiate this class.
*/
private Math() {}
/**
* The {@code double} value that is closer than any other to
* <i>e</i>, the base of the natural logarithms.
*/
public static final double E = 2.718281828459045;
/**
* The {@code double} value that is closer than any other to
* <i>pi</i> (&pi;), the ratio of the circumference of a circle to
* its diameter.
*/
public static final double PI = 3.141592653589793;
/**
* The {@code double} value that is closer than any other to
* <i>tau</i> (&tau;), the ratio of the circumference of a circle
* to its radius.
*
* @apiNote
* The value of <i>pi</i> is one half that of <i>tau</i>; in other
* words, <i>tau</i> is double <i>pi</i> .
*
* @since 19
*/
public static final double TAU = 2.0 * PI;
/**
* Constant by which to multiply an angular value in degrees to obtain an
* angular value in radians.
*/
private static final double DEGREES_TO_RADIANS = 0.017453292519943295;
/**
* Constant by which to multiply an angular value in radians to obtain an
* angular value in degrees.
*/
private static final double RADIANS_TO_DEGREES = 57.29577951308232;
/**
* Returns the trigonometric sine of an angle. Special cases:
* <ul><li>If the argument is NaN or an infinity, then the
* result is NaN.
* <li>If the argument is zero, then the result is a zero with the
* same sign as the argument.</ul>
*
* <p>The computed result must be within 1 ulp of the exact result.
* Results must be semi-monotonic.
*
* @param a an angle, in radians.
* @return the sine of the argument.
*/
@IntrinsicCandidate
public static double sin(double a) {
return StrictMath.sin(a); // default impl. delegates to StrictMath
}
/**
* Returns the trigonometric cosine of an angle. Special cases:
* <ul><li>If the argument is NaN or an infinity, then the
* result is NaN.
* <li>If the argument is zero, then the result is {@code 1.0}.
*</ul>
*
* <p>The computed result must be within 1 ulp of the exact result.
* Results must be semi-monotonic.
*
* @param a an angle, in radians.
* @return the cosine of the argument.
*/
@IntrinsicCandidate
public static double cos(double a) {
return StrictMath.cos(a); // default impl. delegates to StrictMath
}
/**
* Returns the trigonometric tangent of an angle. Special cases:
* <ul><li>If the argument is NaN or an infinity, then the result
* is NaN.
* <li>If the argument is zero, then the result is a zero with the
* same sign as the argument.</ul>
*
* <p>The computed result must be within 1.25 ulps of the exact result.
* Results must be semi-monotonic.
*
* @param a an angle, in radians.
* @return the tangent of the argument.
*/
@IntrinsicCandidate
public static double tan(double a) {
return StrictMath.tan(a); // default impl. delegates to StrictMath
}
/**
* Returns the arc sine of a value; the returned angle is in the
* range &minus;<i>pi</i>/2 through <i>pi</i>/2. Special cases:
* <ul><li>If the argument is NaN or its absolute value is greater
* than 1, then the result is NaN.
* <li>If the argument is zero, then the result is a zero with the
* same sign as the argument.</ul>
*
* <p>The computed result must be within 1 ulp of the exact result.
* Results must be semi-monotonic.
*
* @param a the value whose arc sine is to be returned.
* @return the arc sine of the argument.
*/
public static double asin(double a) {
return StrictMath.asin(a); // default impl. delegates to StrictMath
}
/**
* Returns the arc cosine of a value; the returned angle is in the
* range 0.0 through <i>pi</i>. Special case:
* <ul><li>If the argument is NaN or its absolute value is greater
* than 1, then the result is NaN.
* <li>If the argument is {@code 1.0}, the result is positive zero.
* </ul>
*
* <p>The computed result must be within 1 ulp of the exact result.
* Results must be semi-monotonic.
*
* @param a the value whose arc cosine is to be returned.
* @return the arc cosine of the argument.
*/
public static double acos(double a) {
return StrictMath.acos(a); // default impl. delegates to StrictMath
}
/**
* Returns the arc tangent of a value; the returned angle is in the
* range &minus;<i>pi</i>/2 through <i>pi</i>/2. Special cases:
* <ul><li>If the argument is NaN, then the result is NaN.
* <li>If the argument is zero, then the result is a zero with the
* same sign as the argument.
* <li>If the argument is {@linkplain Double#isInfinite infinite},
* then the result is the closest value to <i>pi</i>/2 with the
* same sign as the input.
* </ul>
*
* <p>The computed result must be within 1 ulp of the exact result.
* Results must be semi-monotonic.
*
* @param a the value whose arc tangent is to be returned.
* @return the arc tangent of the argument.
*/
public static double atan(double a) {
return StrictMath.atan(a); // default impl. delegates to StrictMath
}
/**
* Converts an angle measured in degrees to an approximately
* equivalent angle measured in radians. The conversion from
* degrees to radians is generally inexact.
*
* @param angdeg an angle, in degrees
* @return the measurement of the angle {@code angdeg}
* in radians.
* @since 1.2
*/
public static double toRadians(double angdeg) {
return angdeg * DEGREES_TO_RADIANS;
}
/**
* Converts an angle measured in radians to an approximately
* equivalent angle measured in degrees. The conversion from
* radians to degrees is generally inexact; users should
* <i>not</i> expect {@code cos(toRadians(90.0))} to exactly
* equal {@code 0.0}.
*
* @param angrad an angle, in radians
* @return the measurement of the angle {@code angrad}
* in degrees.
* @since 1.2
*/
public static double toDegrees(double angrad) {
return angrad * RADIANS_TO_DEGREES;
}
/**
* Returns Euler's number <i>e</i> raised to the power of a
* {@code double} value. Special cases:
* <ul><li>If the argument is NaN, the result is NaN.
* <li>If the argument is positive infinity, then the result is
* positive infinity.
* <li>If the argument is negative infinity, then the result is
* positive zero.
* <li>If the argument is zero, then the result is {@code 1.0}.
* </ul>
*
* <p>The computed result must be within 1 ulp of the exact result.
* Results must be semi-monotonic.
*
* @param a the exponent to raise <i>e</i> to.
* @return the value <i>e</i><sup>{@code a}</sup>,
* where <i>e</i> is the base of the natural logarithms.
*/
@IntrinsicCandidate
public static double exp(double a) {
return StrictMath.exp(a); // default impl. delegates to StrictMath
}
/**
* Returns the natural logarithm (base <i>e</i>) of a {@code double}
* value. Special cases:
* <ul><li>If the argument is NaN or less than zero, then the result
* is NaN.
* <li>If the argument is positive infinity, then the result is
* positive infinity.
* <li>If the argument is positive zero or negative zero, then the
* result is negative infinity.
* <li>If the argument is {@code 1.0}, then the result is positive
* zero.
* </ul>
*
* <p>The computed result must be within 1 ulp of the exact result.
* Results must be semi-monotonic.
*
* @param a a value
* @return the value ln&nbsp;{@code a}, the natural logarithm of
* {@code a}.
*/
@IntrinsicCandidate
public static double log(double a) {
return StrictMath.log(a); // default impl. delegates to StrictMath
}
/**
* Returns the base 10 logarithm of a {@code double} value.
* Special cases:
*
* <ul><li>If the argument is NaN or less than zero, then the result
* is NaN.
* <li>If the argument is positive infinity, then the result is
* positive infinity.
* <li>If the argument is positive zero or negative zero, then the
* result is negative infinity.
* <li>If the argument is equal to 10<sup><i>n</i></sup> for
* integer <i>n</i>, then the result is <i>n</i>. In particular,
* if the argument is {@code 1.0} (10<sup>0</sup>), then the
* result is positive zero.
* </ul>
*
* <p>The computed result must be within 1 ulp of the exact result.
* Results must be semi-monotonic.
*
* @param a a value
* @return the base 10 logarithm of {@code a}.
* @since 1.5
*/
@IntrinsicCandidate
public static double log10(double a) {
return StrictMath.log10(a); // default impl. delegates to StrictMath
}
/**
* Returns the correctly rounded positive square root of a
* {@code double} value.
* Special cases:
* <ul><li>If the argument is NaN or less than zero, then the result
* is NaN.
* <li>If the argument is positive infinity, then the result is positive
* infinity.
* <li>If the argument is positive zero or negative zero, then the
* result is the same as the argument.</ul>
* Otherwise, the result is the {@code double} value closest to
* the true mathematical square root of the argument value.
*
* @apiNote
* This method corresponds to the squareRoot operation defined in
* IEEE 754.
*
* @param a a value.
* @return the positive square root of {@code a}.
* If the argument is NaN or less than zero, the result is NaN.
*/
@IntrinsicCandidate
public static double sqrt(double a) {
return StrictMath.sqrt(a); // default impl. delegates to StrictMath
// Note that hardware sqrt instructions
// frequently can be directly used by JITs
// and should be much faster than doing
// Math.sqrt in software.
}
/**
* Returns the cube root of a {@code double} value. For
* positive finite {@code x}, {@code cbrt(-x) ==
* -cbrt(x)}; that is, the cube root of a negative value is
* the negative of the cube root of that value's magnitude.
*
* Special cases:
*
* <ul>
*
* <li>If the argument is NaN, then the result is NaN.
*
* <li>If the argument is infinite, then the result is an infinity
* with the same sign as the argument.
*
* <li>If the argument is zero, then the result is a zero with the
* same sign as the argument.
*
* </ul>
*
* <p>The computed result must be within 1 ulp of the exact result.
*
* @param a a value.
* @return the cube root of {@code a}.
* @since 1.5
*/
@IntrinsicCandidate
public static double cbrt(double a) {
return StrictMath.cbrt(a);
}
/**
* Computes the remainder operation on two arguments as prescribed
* by the IEEE 754 standard.
* The remainder value is mathematically equal to
* <code>f1&nbsp;-&nbsp;f2</code>&nbsp;&times;&nbsp;<i>n</i>,
* where <i>n</i> is the mathematical integer closest to the exact
* mathematical value of the quotient {@code f1/f2}, and if two
* mathematical integers are equally close to {@code f1/f2},
* then <i>n</i> is the integer that is even. If the remainder is
* zero, its sign is the same as the sign of the first argument.
* Special cases:
* <ul><li>If either argument is NaN, or the first argument is infinite,
* or the second argument is positive zero or negative zero, then the
* result is NaN.
* <li>If the first argument is finite and the second argument is
* infinite, then the result is the same as the first argument.</ul>
*
* @param f1 the dividend.
* @param f2 the divisor.
* @return the remainder when {@code f1} is divided by
* {@code f2}.
*/
public static double IEEEremainder(double f1, double f2) {
return StrictMath.IEEEremainder(f1, f2); // delegate to StrictMath
}
/**
* Returns the smallest (closest to negative infinity)
* {@code double} value that is greater than or equal to the
* argument and is equal to a mathematical integer. Special cases:
* <ul><li>If the argument value is already equal to a
* mathematical integer, then the result is the same as the
* argument. <li>If the argument is NaN or an infinity or
* positive zero or negative zero, then the result is the same as
* the argument. <li>If the argument value is less than zero but
* greater than -1.0, then the result is negative zero.</ul> Note
* that the value of {@code Math.ceil(x)} is exactly the
* value of {@code -Math.floor(-x)}.
*
* @apiNote
* This method corresponds to the roundToIntegralTowardPositive
* operation defined in IEEE 754.
*
* @param a a value.
* @return the smallest (closest to negative infinity)
* floating-point value that is greater than or equal to
* the argument and is equal to a mathematical integer.
*/
@IntrinsicCandidate
public static double ceil(double a) {
return StrictMath.ceil(a); // default impl. delegates to StrictMath
}
/**
* Returns the largest (closest to positive infinity)
* {@code double} value that is less than or equal to the
* argument and is equal to a mathematical integer. Special cases:
* <ul><li>If the argument value is already equal to a
* mathematical integer, then the result is the same as the
* argument. <li>If the argument is NaN or an infinity or
* positive zero or negative zero, then the result is the same as
* the argument.</ul>
*
* @apiNote
* This method corresponds to the roundToIntegralTowardNegative
* operation defined in IEEE 754.
*
* @param a a value.
* @return the largest (closest to positive infinity)
* floating-point value that less than or equal to the argument
* and is equal to a mathematical integer.
*/
@IntrinsicCandidate
public static double floor(double a) {
return StrictMath.floor(a); // default impl. delegates to StrictMath
}
/**
* Returns the {@code double} value that is closest in value
* to the argument and is equal to a mathematical integer. If two
* {@code double} values that are mathematical integers are
* equally close, the result is the integer value that is
* even. Special cases:
* <ul><li>If the argument value is already equal to a mathematical
* integer, then the result is the same as the argument.
* <li>If the argument is NaN or an infinity or positive zero or negative
* zero, then the result is the same as the argument.</ul>
*
* @apiNote
* This method corresponds to the roundToIntegralTiesToEven
* operation defined in IEEE 754.
*
* @param a a {@code double} value.
* @return the closest floating-point value to {@code a} that is
* equal to a mathematical integer.
*/
@IntrinsicCandidate
public static double rint(double a) {
return StrictMath.rint(a); // default impl. delegates to StrictMath
}
/**
* Returns the angle <i>theta</i> from the conversion of rectangular
* coordinates ({@code x},&nbsp;{@code y}) to polar
* coordinates (r,&nbsp;<i>theta</i>).
* This method computes the phase <i>theta</i> by computing an arc tangent
* of {@code y/x} in the range of &minus;<i>pi</i> to <i>pi</i>. Special
* cases:
* <ul><li>If either argument is NaN, then the result is NaN.
* <li>If the first argument is positive zero and the second argument
* is positive, or the first argument is positive and finite and the
* second argument is positive infinity, then the result is positive
* zero.
* <li>If the first argument is negative zero and the second argument
* is positive, or the first argument is negative and finite and the
* second argument is positive infinity, then the result is negative zero.
* <li>If the first argument is positive zero and the second argument
* is negative, or the first argument is positive and finite and the
* second argument is negative infinity, then the result is the
* {@code double} value closest to <i>pi</i>.
* <li>If the first argument is negative zero and the second argument
* is negative, or the first argument is negative and finite and the
* second argument is negative infinity, then the result is the
* {@code double} value closest to -<i>pi</i>.
* <li>If the first argument is positive and the second argument is
* positive zero or negative zero, or the first argument is positive
* infinity and the second argument is finite, then the result is the
* {@code double} value closest to <i>pi</i>/2.
* <li>If the first argument is negative and the second argument is
* positive zero or negative zero, or the first argument is negative
* infinity and the second argument is finite, then the result is the
* {@code double} value closest to -<i>pi</i>/2.
* <li>If both arguments are positive infinity, then the result is the
* {@code double} value closest to <i>pi</i>/4.
* <li>If the first argument is positive infinity and the second argument
* is negative infinity, then the result is the {@code double}
* value closest to 3*<i>pi</i>/4.
* <li>If the first argument is negative infinity and the second argument
* is positive infinity, then the result is the {@code double} value
* closest to -<i>pi</i>/4.
* <li>If both arguments are negative infinity, then the result is the
* {@code double} value closest to -3*<i>pi</i>/4.</ul>
*
* <p>The computed result must be within 2 ulps of the exact result.
* Results must be semi-monotonic.
*
* @apiNote
* For <i>y</i> with a positive sign and finite nonzero
* <i>x</i>, the exact mathematical value of {@code atan2} is
* equal to:
* <ul>
* <li>If <i>x</i> {@literal >} 0, atan(abs(<i>y</i>/<i>x</i>))
* <li>If <i>x</i> {@literal <} 0, &pi; - atan(abs(<i>y</i>/<i>x</i>))
* </ul>
*
* @param y the ordinate coordinate
* @param x the abscissa coordinate
* @return the <i>theta</i> component of the point
* (<i>r</i>,&nbsp;<i>theta</i>)
* in polar coordinates that corresponds to the point
* (<i>x</i>,&nbsp;<i>y</i>) in Cartesian coordinates.
*/
@IntrinsicCandidate
public static double atan2(double y, double x) {
return StrictMath.atan2(y, x); // default impl. delegates to StrictMath
}
/**
* Returns the value of the first argument raised to the power of the
* second argument. Special cases:
*
* <ul><li>If the second argument is positive or negative zero, then the
* result is 1.0.
* <li>If the second argument is 1.0, then the result is the same as the
* first argument.
* <li>If the second argument is NaN, then the result is NaN.
* <li>If the first argument is NaN and the second argument is nonzero,
* then the result is NaN.
*
* <li>If
* <ul>
* <li>the absolute value of the first argument is greater than 1
* and the second argument is positive infinity, or
* <li>the absolute value of the first argument is less than 1 and
* the second argument is negative infinity,
* </ul>
* then the result is positive infinity.
*
* <li>If
* <ul>
* <li>the absolute value of the first argument is greater than 1 and
* the second argument is negative infinity, or
* <li>the absolute value of the
* first argument is less than 1 and the second argument is positive
* infinity,
* </ul>
* then the result is positive zero.
*
* <li>If the absolute value of the first argument equals 1 and the
* second argument is infinite, then the result is NaN.
*
* <li>If
* <ul>
* <li>the first argument is positive zero and the second argument
* is greater than zero, or
* <li>the first argument is positive infinity and the second
* argument is less than zero,
* </ul>
* then the result is positive zero.
*
* <li>If
* <ul>
* <li>the first argument is positive zero and the second argument
* is less than zero, or
* <li>the first argument is positive infinity and the second
* argument is greater than zero,
* </ul>
* then the result is positive infinity.
*
* <li>If
* <ul>
* <li>the first argument is negative zero and the second argument
* is greater than zero but not a finite odd integer, or
* <li>the first argument is negative infinity and the second
* argument is less than zero but not a finite odd integer,
* </ul>
* then the result is positive zero.
*
* <li>If
* <ul>
* <li>the first argument is negative zero and the second argument
* is a positive finite odd integer, or
* <li>the first argument is negative infinity and the second
* argument is a negative finite odd integer,
* </ul>
* then the result is negative zero.
*
* <li>If
* <ul>
* <li>the first argument is negative zero and the second argument
* is less than zero but not a finite odd integer, or
* <li>the first argument is negative infinity and the second
* argument is greater than zero but not a finite odd integer,
* </ul>
* then the result is positive infinity.
*
* <li>If
* <ul>
* <li>the first argument is negative zero and the second argument
* is a negative finite odd integer, or
* <li>the first argument is negative infinity and the second
* argument is a positive finite odd integer,
* </ul>
* then the result is negative infinity.
*
* <li>If the first argument is finite and less than zero
* <ul>
* <li> if the second argument is a finite even integer, the
* result is equal to the result of raising the absolute value of
* the first argument to the power of the second argument
*
* <li>if the second argument is a finite odd integer, the result
* is equal to the negative of the result of raising the absolute
* value of the first argument to the power of the second
* argument
*
* <li>if the second argument is finite and not an integer, then
* the result is NaN.
* </ul>
*
* <li>If both arguments are integers, then the result is exactly equal
* to the mathematical result of raising the first argument to the power
* of the second argument if that result can in fact be represented
* exactly as a {@code double} value.</ul>
*
* <p>(In the foregoing descriptions, a floating-point value is
* considered to be an integer if and only if it is finite and a
* fixed point of the method {@link #ceil ceil} or,
* equivalently, a fixed point of the method {@link #floor
* floor}. A value is a fixed point of a one-argument
* method if and only if the result of applying the method to the
* value is equal to the value.)
*
* <p>The computed result must be within 1 ulp of the exact result.
* Results must be semi-monotonic.
*
* @apiNote
* The special cases definitions of this method differ from the
* special case definitions of the IEEE 754 recommended {@code
* pow} operation for &plusmn;{@code 1.0} raised to an infinite
* power. This method treats such cases as indeterminate and
* specifies a NaN is returned. The IEEE 754 specification treats
* the infinite power as a large integer (large-magnitude
* floating-point numbers are numerically integers, specifically
* even integers) and therefore specifies {@code 1.0} be returned.
*
* @param a the base.
* @param b the exponent.
* @return the value {@code a}<sup>{@code b}</sup>.
*/
@IntrinsicCandidate
public static double pow(double a, double b) {
return StrictMath.pow(a, b); // default impl. delegates to StrictMath
}
/**
* Returns the closest {@code int} to the argument, with ties
* rounding to positive infinity.
*
* <p>
* Special cases:
* <ul><li>If the argument is NaN, the result is 0.
* <li>If the argument is negative infinity or any value less than or
* equal to the value of {@code Integer.MIN_VALUE}, the result is
* equal to the value of {@code Integer.MIN_VALUE}.
* <li>If the argument is positive infinity or any value greater than or
* equal to the value of {@code Integer.MAX_VALUE}, the result is
* equal to the value of {@code Integer.MAX_VALUE}.</ul>
*
* @param a a floating-point value to be rounded to an integer.
* @return the value of the argument rounded to the nearest
* {@code int} value.
* @see java.lang.Integer#MAX_VALUE
* @see java.lang.Integer#MIN_VALUE
*/
@IntrinsicCandidate
public static int round(float a) {
int intBits = Float.floatToRawIntBits(a);
int biasedExp = (intBits & FloatConsts.EXP_BIT_MASK)
>> (FloatConsts.SIGNIFICAND_WIDTH - 1);
int shift = (FloatConsts.SIGNIFICAND_WIDTH - 2
+ FloatConsts.EXP_BIAS) - biasedExp;
if ((shift & -32) == 0) { // shift >= 0 && shift < 32
// a is a finite number such that pow(2,-32) <= ulp(a) < 1
int r = ((intBits & FloatConsts.SIGNIF_BIT_MASK)
| (FloatConsts.SIGNIF_BIT_MASK + 1));
if (intBits < 0) {
r = -r;
}
// In the comments below each Java expression evaluates to the value
// the corresponding mathematical expression:
// (r) evaluates to a / ulp(a)
// (r >> shift) evaluates to floor(a * 2)
// ((r >> shift) + 1) evaluates to floor((a + 1/2) * 2)
// (((r >> shift) + 1) >> 1) evaluates to floor(a + 1/2)
return ((r >> shift) + 1) >> 1;
} else {
// a is either
// - a finite number with abs(a) < exp(2,FloatConsts.SIGNIFICAND_WIDTH-32) < 1/2
// - a finite number with ulp(a) >= 1 and hence a is a mathematical integer
// - an infinity or NaN
return (int) a;
}
}
/**
* Returns the closest {@code long} to the argument, with ties
* rounding to positive infinity.
*
* <p>Special cases:
* <ul><li>If the argument is NaN, the result is 0.
* <li>If the argument is negative infinity or any value less than or
* equal to the value of {@code Long.MIN_VALUE}, the result is
* equal to the value of {@code Long.MIN_VALUE}.
* <li>If the argument is positive infinity or any value greater than or
* equal to the value of {@code Long.MAX_VALUE}, the result is
* equal to the value of {@code Long.MAX_VALUE}.</ul>
*
* @param a a floating-point value to be rounded to a
* {@code long}.
* @return the value of the argument rounded to the nearest
* {@code long} value.
* @see java.lang.Long#MAX_VALUE
* @see java.lang.Long#MIN_VALUE
*/
@IntrinsicCandidate
public static long round(double a) {
long longBits = Double.doubleToRawLongBits(a);
long biasedExp = (longBits & DoubleConsts.EXP_BIT_MASK)
>> (DoubleConsts.SIGNIFICAND_WIDTH - 1);
long shift = (DoubleConsts.SIGNIFICAND_WIDTH - 2
+ DoubleConsts.EXP_BIAS) - biasedExp;
if ((shift & -64) == 0) { // shift >= 0 && shift < 64
// a is a finite number such that pow(2,-64) <= ulp(a) < 1
long r = ((longBits & DoubleConsts.SIGNIF_BIT_MASK)
| (DoubleConsts.SIGNIF_BIT_MASK + 1));
if (longBits < 0) {
r = -r;
}
// In the comments below each Java expression evaluates to the value
// the corresponding mathematical expression:
// (r) evaluates to a / ulp(a)
// (r >> shift) evaluates to floor(a * 2)
// ((r >> shift) + 1) evaluates to floor((a + 1/2) * 2)
// (((r >> shift) + 1) >> 1) evaluates to floor(a + 1/2)
return ((r >> shift) + 1) >> 1;
} else {
// a is either
// - a finite number with abs(a) < exp(2,DoubleConsts.SIGNIFICAND_WIDTH-64) < 1/2
// - a finite number with ulp(a) >= 1 and hence a is a mathematical integer
// - an infinity or NaN
return (long) a;
}
}
private static final class RandomNumberGeneratorHolder {
static final Random randomNumberGenerator = new Random();
}
/**
* Returns a {@code double} value with a positive sign, greater
* than or equal to {@code 0.0} and less than {@code 1.0}.
* Returned values are chosen pseudorandomly with (approximately)
* uniform distribution from that range.
*
* <p>When this method is first called, it creates a single new
* pseudorandom-number generator, exactly as if by the expression
*
* <blockquote>{@code new java.util.Random()}</blockquote>
*
* This new pseudorandom-number generator is used thereafter for
* all calls to this method and is used nowhere else.
*
* <p>This method is properly synchronized to allow correct use by
* more than one thread. However, if many threads need to generate
* pseudorandom numbers at a great rate, it may reduce contention
* for each thread to have its own pseudorandom-number generator.
*
* @apiNote
* As the largest {@code double} value less than {@code 1.0}
* is {@code Math.nextDown(1.0)}, a value {@code x} in the closed range
* {@code [x1,x2]} where {@code x1<=x2} may be defined by the statements
*
* <blockquote><pre>{@code
* double f = Math.random()/Math.nextDown(1.0);
* double x = x1*(1.0 - f) + x2*f;
* }</pre></blockquote>
*
* @return a pseudorandom {@code double} greater than or equal
* to {@code 0.0} and less than {@code 1.0}.
* @see #nextDown(double)
* @see Random#nextDouble()
*/
public static double random() {
return RandomNumberGeneratorHolder.randomNumberGenerator.nextDouble();
}
/**
* Returns the sum of its arguments,
* throwing an exception if the result overflows an {@code int}.
*
* @param x the first value
* @param y the second value
* @return the result
* @throws ArithmeticException if the result overflows an int
* @since 1.8
*/
@IntrinsicCandidate
public static int addExact(int x, int y) {
int r = x + y;
// HD 2-12 Overflow iff both arguments have the opposite sign of the result
if (((x ^ r) & (y ^ r)) < 0) {
throw new ArithmeticException("integer overflow");
}
return r;
}
/**
* Returns the sum of its arguments,
* throwing an exception if the result overflows a {@code long}.
*
* @param x the first value
* @param y the second value
* @return the result
* @throws ArithmeticException if the result overflows a long
* @since 1.8
*/
@IntrinsicCandidate
public static long addExact(long x, long y) {
long r = x + y;
// HD 2-12 Overflow iff both arguments have the opposite sign of the result
if (((x ^ r) & (y ^ r)) < 0) {
throw new ArithmeticException("long overflow");
}
return r;
}
/**
* Returns the difference of the arguments,
* throwing an exception if the result overflows an {@code int}.
*
* @param x the first value
* @param y the second value to subtract from the first
* @return the result
* @throws ArithmeticException if the result overflows an int
* @since 1.8
*/
@IntrinsicCandidate
public static int subtractExact(int x, int y) {
int r = x - y;
// HD 2-12 Overflow iff the arguments have different signs and
// the sign of the result is different from the sign of x
if (((x ^ y) & (x ^ r)) < 0) {
throw new ArithmeticException("integer overflow");
}
return r;
}
/**
* Returns the difference of the arguments,
* throwing an exception if the result overflows a {@code long}.
*
* @param x the first value
* @param y the second value to subtract from the first
* @return the result
* @throws ArithmeticException if the result overflows a long
* @since 1.8
*/
@IntrinsicCandidate
public static long subtractExact(long x, long y) {
long r = x - y;
// HD 2-12 Overflow iff the arguments have different signs and
// the sign of the result is different from the sign of x
if (((x ^ y) & (x ^ r)) < 0) {
throw new ArithmeticException("long overflow");
}
return r;
}
/**
* Returns the product of the arguments,
* throwing an exception if the result overflows an {@code int}.
*
* @param x the first value
* @param y the second value
* @return the result
* @throws ArithmeticException if the result overflows an int
* @since 1.8
*/
@IntrinsicCandidate
public static int multiplyExact(int x, int y) {
long r = (long)x * (long)y;
if ((int)r != r) {
throw new ArithmeticException("integer overflow");
}
return (int)r;
}
/**
* Returns the product of the arguments, throwing an exception if the result
* overflows a {@code long}.
*
* @param x the first value
* @param y the second value
* @return the result
* @throws ArithmeticException if the result overflows a long
* @since 9
*/
public static long multiplyExact(long x, int y) {
return multiplyExact(x, (long)y);
}
/**
* Returns the product of the arguments,
* throwing an exception if the result overflows a {@code long}.
*
* @param x the first value
* @param y the second value
* @return the result
* @throws ArithmeticException if the result overflows a long
* @since 1.8
*/
@IntrinsicCandidate
public static long multiplyExact(long x, long y) {
long r = x * y;
long ax = Math.abs(x);
long ay = Math.abs(y);
if (((ax | ay) >>> 31 != 0)) {
// Some bits greater than 2^31 that might cause overflow
// Check the result using the divide operator
// and check for the special case of Long.MIN_VALUE * -1
if (((y != 0) && (r / y != x)) ||
(x == Long.MIN_VALUE && y == -1)) {
throw new ArithmeticException("long overflow");
}
}
return r;
}
/**
* Returns the quotient of the arguments, throwing an exception if the
* result overflows an {@code int}. Such overflow occurs in this method if
* {@code x} is {@link Integer#MIN_VALUE} and {@code y} is {@code -1}.
* In contrast, if {@code Integer.MIN_VALUE / -1} were evaluated directly,
* the result would be {@code Integer.MIN_VALUE} and no exception would be
* thrown.
* <p>
* If {@code y} is zero, an {@code ArithmeticException} is thrown
* (JLS {@jls 15.17.2}).
* <p>
* The built-in remainder operator "{@code %}" is a suitable counterpart
* both for this method and for the built-in division operator "{@code /}".
*
* @param x the dividend
* @param y the divisor
* @return the quotient {@code x / y}
* @throws ArithmeticException if {@code y} is zero or the quotient
* overflows an int
* @jls 15.17.2 Division Operator /
* @since 18
*/
public static int divideExact(int x, int y) {
int q = x / y;
if ((x & y & q) >= 0) {
return q;
}
throw new ArithmeticException("integer overflow");
}
/**
* Returns the quotient of the arguments, throwing an exception if the
* result overflows a {@code long}. Such overflow occurs in this method if
* {@code x} is {@link Long#MIN_VALUE} and {@code y} is {@code -1}.
* In contrast, if {@code Long.MIN_VALUE / -1} were evaluated directly,
* the result would be {@code Long.MIN_VALUE} and no exception would be
* thrown.
* <p>
* If {@code y} is zero, an {@code ArithmeticException} is thrown
* (JLS {@jls 15.17.2}).
* <p>
* The built-in remainder operator "{@code %}" is a suitable counterpart
* both for this method and for the built-in division operator "{@code /}".
*
* @param x the dividend
* @param y the divisor
* @return the quotient {@code x / y}
* @throws ArithmeticException if {@code y} is zero or the quotient
* overflows a long
* @jls 15.17.2 Division Operator /
* @since 18
*/
public static long divideExact(long x, long y) {
long q = x / y;
if ((x & y & q) >= 0) {
return q;
}
throw new ArithmeticException("long overflow");
}
/**
* Returns the largest (closest to positive infinity)
* {@code int} value that is less than or equal to the algebraic quotient.
* This method is identical to {@link #floorDiv(int,int)} except that it
* throws an {@code ArithmeticException} when the dividend is
* {@linkplain Integer#MIN_VALUE Integer.MIN_VALUE} and the divisor is
* {@code -1} instead of ignoring the integer overflow and returning
* {@code Integer.MIN_VALUE}.
* <p>
* The floor modulus method {@link #floorMod(int,int)} is a suitable
* counterpart both for this method and for the {@link #floorDiv(int,int)}
* method.
* <p>
* For examples, see {@link #floorDiv(int, int)}.
*
* @param x the dividend
* @param y the divisor
* @return the largest (closest to positive infinity)
* {@code int} value that is less than or equal to the algebraic quotient.
* @throws ArithmeticException if the divisor {@code y} is zero, or the
* dividend {@code x} is {@code Integer.MIN_VALUE} and the divisor {@code y}
* is {@code -1}.
* @see #floorDiv(int, int)
* @since 18
*/
public static int floorDivExact(int x, int y) {
final int q = x / y;
if ((x & y & q) >= 0) {
// if the signs are different and modulo not zero, round down
if ((x ^ y) < 0 && (q * y != x)) {
return q - 1;
}
return q;
}
throw new ArithmeticException("integer overflow");
}
/**
* Returns the largest (closest to positive infinity)
* {@code long} value that is less than or equal to the algebraic quotient.
* This method is identical to {@link #floorDiv(long,long)} except that it
* throws an {@code ArithmeticException} when the dividend is
* {@linkplain Long#MIN_VALUE Long.MIN_VALUE} and the divisor is
* {@code -1} instead of ignoring the integer overflow and returning
* {@code Long.MIN_VALUE}.
* <p>
* The floor modulus method {@link #floorMod(long,long)} is a suitable
* counterpart both for this method and for the {@link #floorDiv(long,long)}
* method.
* <p>
* For examples, see {@link #floorDiv(int, int)}.
*
* @param x the dividend
* @param y the divisor
* @return the largest (closest to positive infinity)
* {@code long} value that is less than or equal to the algebraic quotient.
* @throws ArithmeticException if the divisor {@code y} is zero, or the
* dividend {@code x} is {@code Long.MIN_VALUE} and the divisor {@code y}
* is {@code -1}.
* @see #floorDiv(long,long)
* @since 18
*/
public static long floorDivExact(long x, long y) {
final long q = x / y;
if ((x & y & q) >= 0) {
// if the signs are different and modulo not zero, round down
if ((x ^ y) < 0 && (q * y != x)) {
return q - 1;
}
return q;
}
throw new ArithmeticException("long overflow");
}
/**
* Returns the smallest (closest to negative infinity)
* {@code int} value that is greater than or equal to the algebraic quotient.
* This method is identical to {@link #ceilDiv(int,int)} except that it
* throws an {@code ArithmeticException} when the dividend is
* {@linkplain Integer#MIN_VALUE Integer.MIN_VALUE} and the divisor is
* {@code -1} instead of ignoring the integer overflow and returning
* {@code Integer.MIN_VALUE}.
* <p>
* The ceil modulus method {@link #ceilMod(int,int)} is a suitable
* counterpart both for this method and for the {@link #ceilDiv(int,int)}
* method.
* <p>
* For examples, see {@link #ceilDiv(int, int)}.
*
* @param x the dividend
* @param y the divisor
* @return the smallest (closest to negative infinity)
* {@code int} value that is greater than or equal to the algebraic quotient.
* @throws ArithmeticException if the divisor {@code y} is zero, or the
* dividend {@code x} is {@code Integer.MIN_VALUE} and the divisor {@code y}
* is {@code -1}.
* @see #ceilDiv(int, int)
* @since 18
*/
public static int ceilDivExact(int x, int y) {
final int q = x / y;
if ((x & y & q) >= 0) {
// if the signs are the same and modulo not zero, round up
if ((x ^ y) >= 0 && (q * y != x)) {
return q + 1;
}
return q;
}
throw new ArithmeticException("integer overflow");
}
/**
* Returns the smallest (closest to negative infinity)
* {@code long} value that is greater than or equal to the algebraic quotient.
* This method is identical to {@link #ceilDiv(long,long)} except that it
* throws an {@code ArithmeticException} when the dividend is
* {@linkplain Long#MIN_VALUE Long.MIN_VALUE} and the divisor is
* {@code -1} instead of ignoring the integer overflow and returning
* {@code Long.MIN_VALUE}.
* <p>
* The ceil modulus method {@link #ceilMod(long,long)} is a suitable
* counterpart both for this method and for the {@link #ceilDiv(long,long)}
* method.
* <p>
* For examples, see {@link #ceilDiv(int, int)}.
*
* @param x the dividend
* @param y the divisor
* @return the smallest (closest to negative infinity)
* {@code long} value that is greater than or equal to the algebraic quotient.
* @throws ArithmeticException if the divisor {@code y} is zero, or the
* dividend {@code x} is {@code Long.MIN_VALUE} and the divisor {@code y}
* is {@code -1}.
* @see #ceilDiv(long,long)
* @since 18
*/
public static long ceilDivExact(long x, long y) {
final long q = x / y;
if ((x & y & q) >= 0) {
// if the signs are the same and modulo not zero, round up
if ((x ^ y) >= 0 && (q * y != x)) {
return q + 1;
}
return q;
}
throw new ArithmeticException("long overflow");
}
/**
* Returns the argument incremented by one, throwing an exception if the
* result overflows an {@code int}.
* The overflow only occurs for {@linkplain Integer#MAX_VALUE the maximum value}.
*
* @param a the value to increment
* @return the result
* @throws ArithmeticException if the result overflows an int
* @since 1.8
*/
@IntrinsicCandidate
public static int incrementExact(int a) {
if (a == Integer.MAX_VALUE) {
throw new ArithmeticException("integer overflow");
}
return a + 1;
}
/**
* Returns the argument incremented by one, throwing an exception if the
* result overflows a {@code long}.
* The overflow only occurs for {@linkplain Long#MAX_VALUE the maximum value}.
*
* @param a the value to increment
* @return the result
* @throws ArithmeticException if the result overflows a long
* @since 1.8
*/
@IntrinsicCandidate
public static long incrementExact(long a) {
if (a == Long.MAX_VALUE) {
throw new ArithmeticException("long overflow");
}
return a + 1L;
}
/**
* Returns the argument decremented by one, throwing an exception if the
* result overflows an {@code int}.
* The overflow only occurs for {@linkplain Integer#MIN_VALUE the minimum value}.
*
* @param a the value to decrement
* @return the result
* @throws ArithmeticException if the result overflows an int
* @since 1.8
*/
@IntrinsicCandidate
public static int decrementExact(int a) {
if (a == Integer.MIN_VALUE) {
throw new ArithmeticException("integer overflow");
}
return a - 1;
}
/**
* Returns the argument decremented by one, throwing an exception if the
* result overflows a {@code long}.
* The overflow only occurs for {@linkplain Long#MIN_VALUE the minimum value}.
*
* @param a the value to decrement
* @return the result
* @throws ArithmeticException if the result overflows a long
* @since 1.8
*/
@IntrinsicCandidate
public static long decrementExact(long a) {
if (a == Long.MIN_VALUE) {
throw new ArithmeticException("long overflow");
}
return a - 1L;
}
/**
* Returns the negation of the argument, throwing an exception if the
* result overflows an {@code int}.
* The overflow only occurs for {@linkplain Integer#MIN_VALUE the minimum value}.
*
* @param a the value to negate
* @return the result
* @throws ArithmeticException if the result overflows an int
* @since 1.8
*/
@IntrinsicCandidate
public static int negateExact(int a) {
if (a == Integer.MIN_VALUE) {
throw new ArithmeticException("integer overflow");
}
return -a;
}
/**
* Returns the negation of the argument, throwing an exception if the
* result overflows a {@code long}.
* The overflow only occurs for {@linkplain Long#MIN_VALUE the minimum value}.
*
* @param a the value to negate
* @return the result
* @throws ArithmeticException if the result overflows a long
* @since 1.8
*/
@IntrinsicCandidate
public static long negateExact(long a) {
if (a == Long.MIN_VALUE) {
throw new ArithmeticException("long overflow");
}
return -a;
}
/**
* Returns the value of the {@code long} argument,
* throwing an exception if the value overflows an {@code int}.
*
* @param value the long value
* @return the argument as an int
* @throws ArithmeticException if the {@code argument} overflows an int
* @since 1.8
*/
public static int toIntExact(long value) {
if ((int)value != value) {
throw new ArithmeticException("integer overflow");
}
return (int)value;
}
/**
* Returns the exact mathematical product of the arguments.
*
* @param x the first value
* @param y the second value
* @return the result
* @since 9
*/
public static long multiplyFull(int x, int y) {
return (long)x * (long)y;
}
/**
* Returns as a {@code long} the most significant 64 bits of the 128-bit
* product of two 64-bit factors.
*
* @param x the first value
* @param y the second value
* @return the result
* @see #unsignedMultiplyHigh
* @since 9
*/
@IntrinsicCandidate
public static long multiplyHigh(long x, long y) {
// Use technique from section 8-2 of Henry S. Warren, Jr.,
// Hacker's Delight (2nd ed.) (Addison Wesley, 2013), 173-174.
long x1 = x >> 32;
long x2 = x & 0xFFFFFFFFL;
long y1 = y >> 32;
long y2 = y & 0xFFFFFFFFL;
long z2 = x2 * y2;
long t = x1 * y2 + (z2 >>> 32);
long z1 = t & 0xFFFFFFFFL;
long z0 = t >> 32;
z1 += x2 * y1;
return x1 * y1 + z0 + (z1 >> 32);
}
/**
* Returns as a {@code long} the most significant 64 bits of the unsigned
* 128-bit product of two unsigned 64-bit factors.
*
* @param x the first value
* @param y the second value
* @return the result
* @see #multiplyHigh
* @since 18
*/
@IntrinsicCandidate
public static long unsignedMultiplyHigh(long x, long y) {
// Compute via multiplyHigh() to leverage the intrinsic
long result = Math.multiplyHigh(x, y);
result += (y & (x >> 63)); // equivalent to `if (x < 0) result += y;`
result += (x & (y >> 63)); // equivalent to `if (y < 0) result += x;`
return result;
}
/**
* Returns the largest (closest to positive infinity)
* {@code int} value that is less than or equal to the algebraic quotient.
* There is one special case: if the dividend is
* {@linkplain Integer#MIN_VALUE Integer.MIN_VALUE} and the divisor is {@code -1},
* then integer overflow occurs and
* the result is equal to {@code Integer.MIN_VALUE}.
* <p>
* Normal integer division operates under the round to zero rounding mode
* (truncation). This operation instead acts under the round toward
* negative infinity (floor) rounding mode.
* The floor rounding mode gives different results from truncation
* when the exact quotient is not an integer and is negative.
* <ul>
* <li>If the signs of the arguments are the same, the results of
* {@code floorDiv} and the {@code /} operator are the same. <br>
* For example, {@code floorDiv(4, 3) == 1} and {@code (4 / 3) == 1}.</li>
* <li>If the signs of the arguments are different, {@code floorDiv}
* returns the largest integer less than or equal to the quotient
* while the {@code /} operator returns the smallest integer greater
* than or equal to the quotient.
* They differ if and only if the quotient is not an integer.<br>
* For example, {@code floorDiv(-4, 3) == -2},
* whereas {@code (-4 / 3) == -1}.
* </li>
* </ul>
*
* @param x the dividend
* @param y the divisor
* @return the largest (closest to positive infinity)
* {@code int} value that is less than or equal to the algebraic quotient.
* @throws ArithmeticException if the divisor {@code y} is zero
* @see #floorMod(int, int)
* @see #floor(double)
* @since 1.8
*/
public static int floorDiv(int x, int y) {
final int q = x / y;
// if the signs are different and modulo not zero, round down
if ((x ^ y) < 0 && (q * y != x)) {
return q - 1;
}
return q;
}
/**
* Returns the largest (closest to positive infinity)
* {@code long} value that is less than or equal to the algebraic quotient.
* There is one special case: if the dividend is
* {@linkplain Long#MIN_VALUE Long.MIN_VALUE} and the divisor is {@code -1},
* then integer overflow occurs and
* the result is equal to {@code Long.MIN_VALUE}.
* <p>
* Normal integer division operates under the round to zero rounding mode
* (truncation). This operation instead acts under the round toward
* negative infinity (floor) rounding mode.
* The floor rounding mode gives different results from truncation
* when the exact result is not an integer and is negative.
* <p>
* For examples, see {@link #floorDiv(int, int)}.
*
* @param x the dividend
* @param y the divisor
* @return the largest (closest to positive infinity)
* {@code long} value that is less than or equal to the algebraic quotient.
* @throws ArithmeticException if the divisor {@code y} is zero
* @see #floorMod(long, int)
* @see #floor(double)
* @since 9
*/
public static long floorDiv(long x, int y) {
return floorDiv(x, (long)y);
}
/**
* Returns the largest (closest to positive infinity)
* {@code long} value that is less than or equal to the algebraic quotient.
* There is one special case: if the dividend is
* {@linkplain Long#MIN_VALUE Long.MIN_VALUE} and the divisor is {@code -1},
* then integer overflow occurs and
* the result is equal to {@code Long.MIN_VALUE}.
* <p>
* Normal integer division operates under the round to zero rounding mode
* (truncation). This operation instead acts under the round toward
* negative infinity (floor) rounding mode.
* The floor rounding mode gives different results from truncation
* when the exact result is not an integer and is negative.
* <p>
* For examples, see {@link #floorDiv(int, int)}.
*
* @param x the dividend
* @param y the divisor
* @return the largest (closest to positive infinity)
* {@code long} value that is less than or equal to the algebraic quotient.
* @throws ArithmeticException if the divisor {@code y} is zero
* @see #floorMod(long, long)
* @see #floor(double)
* @since 1.8
*/
public static long floorDiv(long x, long y) {
final long q = x / y;
// if the signs are different and modulo not zero, round down
if ((x ^ y) < 0 && (q * y != x)) {
return q - 1;
}
return q;
}
/**
* Returns the floor modulus of the {@code int} arguments.
* <p>
* The floor modulus is {@code r = x - (floorDiv(x, y) * y)},
* has the same sign as the divisor {@code y} or is zero, and
* is in the range of {@code -abs(y) < r < +abs(y)}.
*
* <p>
* The relationship between {@code floorDiv} and {@code floorMod} is such that:
* <ul>
* <li>{@code floorDiv(x, y) * y + floorMod(x, y) == x}</li>
* </ul>
* <p>
* The difference in values between {@code floorMod} and the {@code %} operator
* is due to the difference between {@code floorDiv} and the {@code /}
* operator, as detailed in {@linkplain #floorDiv(int, int)}.
* <p>
* Examples:
* <ul>
* <li>Regardless of the signs of the arguments, {@code floorMod}(x, y)
* is zero exactly when {@code x % y} is zero as well.</li>
* <li>If neither {@code floorMod}(x, y) nor {@code x % y} is zero,
* they differ exactly when the signs of the arguments differ.<br>
* <ul>
* <li>{@code floorMod(+4, +3) == +1}; &nbsp; and {@code (+4 % +3) == +1}</li>
* <li>{@code floorMod(-4, -3) == -1}; &nbsp; and {@code (-4 % -3) == -1}</li>
* <li>{@code floorMod(+4, -3) == -2}; &nbsp; and {@code (+4 % -3) == +1}</li>
* <li>{@code floorMod(-4, +3) == +2}; &nbsp; and {@code (-4 % +3) == -1}</li>
* </ul>
* </li>
* </ul>
*
* @param x the dividend
* @param y the divisor
* @return the floor modulus {@code x - (floorDiv(x, y) * y)}
* @throws ArithmeticException if the divisor {@code y} is zero
* @see #floorDiv(int, int)
* @since 1.8
*/
public static int floorMod(int x, int y) {
final int r = x % y;
// if the signs are different and modulo not zero, adjust result
if ((x ^ y) < 0 && r != 0) {
return r + y;
}
return r;
}
/**
* Returns the floor modulus of the {@code long} and {@code int} arguments.
* <p>
* The floor modulus is {@code r = x - (floorDiv(x, y) * y)},
* has the same sign as the divisor {@code y} or is zero, and
* is in the range of {@code -abs(y) < r < +abs(y)}.
*
* <p>
* The relationship between {@code floorDiv} and {@code floorMod} is such that:
* <ul>
* <li>{@code floorDiv(x, y) * y + floorMod(x, y) == x}</li>
* </ul>
* <p>
* For examples, see {@link #floorMod(int, int)}.
*
* @param x the dividend
* @param y the divisor
* @return the floor modulus {@code x - (floorDiv(x, y) * y)}
* @throws ArithmeticException if the divisor {@code y} is zero
* @see #floorDiv(long, int)
* @since 9
*/
public static int floorMod(long x, int y) {
// Result cannot overflow the range of int.
return (int)floorMod(x, (long)y);
}
/**
* Returns the floor modulus of the {@code long} arguments.
* <p>
* The floor modulus is {@code r = x - (floorDiv(x, y) * y)},
* has the same sign as the divisor {@code y} or is zero, and
* is in the range of {@code -abs(y) < r < +abs(y)}.
*
* <p>
* The relationship between {@code floorDiv} and {@code floorMod} is such that:
* <ul>
* <li>{@code floorDiv(x, y) * y + floorMod(x, y) == x}</li>
* </ul>
* <p>
* For examples, see {@link #floorMod(int, int)}.
*
* @param x the dividend
* @param y the divisor
* @return the floor modulus {@code x - (floorDiv(x, y) * y)}
* @throws ArithmeticException if the divisor {@code y} is zero
* @see #floorDiv(long, long)
* @since 1.8
*/
public static long floorMod(long x, long y) {
final long r = x % y;
// if the signs are different and modulo not zero, adjust result
if ((x ^ y) < 0 && r != 0) {
return r + y;
}
return r;
}
/**
* Returns the smallest (closest to negative infinity)
* {@code int} value that is greater than or equal to the algebraic quotient.
* There is one special case: if the dividend is
* {@linkplain Integer#MIN_VALUE Integer.MIN_VALUE} and the divisor is {@code -1},
* then integer overflow occurs and
* the result is equal to {@code Integer.MIN_VALUE}.
* <p>
* Normal integer division operates under the round to zero rounding mode
* (truncation). This operation instead acts under the round toward
* positive infinity (ceiling) rounding mode.
* The ceiling rounding mode gives different results from truncation
* when the exact quotient is not an integer and is positive.
* <ul>
* <li>If the signs of the arguments are different, the results of
* {@code ceilDiv} and the {@code /} operator are the same. <br>
* For example, {@code ceilDiv(-4, 3) == -1} and {@code (-4 / 3) == -1}.</li>
* <li>If the signs of the arguments are the same, {@code ceilDiv}
* returns the smallest integer greater than or equal to the quotient
* while the {@code /} operator returns the largest integer less
* than or equal to the quotient.
* They differ if and only if the quotient is not an integer.<br>
* For example, {@code ceilDiv(4, 3) == 2},
* whereas {@code (4 / 3) == 1}.
* </li>
* </ul>
*
* @param x the dividend
* @param y the divisor
* @return the smallest (closest to negative infinity)
* {@code int} value that is greater than or equal to the algebraic quotient.
* @throws ArithmeticException if the divisor {@code y} is zero
* @see #ceilMod(int, int)
* @see #ceil(double)
* @since 18
*/
public static int ceilDiv(int x, int y) {
final int q = x / y;
// if the signs are the same and modulo not zero, round up
if ((x ^ y) >= 0 && (q * y != x)) {
return q + 1;
}
return q;
}
/**
* Returns the smallest (closest to negative infinity)
* {@code long} value that is greater than or equal to the algebraic quotient.
* There is one special case: if the dividend is
* {@linkplain Long#MIN_VALUE Long.MIN_VALUE} and the divisor is {@code -1},
* then integer overflow occurs and
* the result is equal to {@code Long.MIN_VALUE}.
* <p>
* Normal integer division operates under the round to zero rounding mode
* (truncation). This operation instead acts under the round toward
* positive infinity (ceiling) rounding mode.
* The ceiling rounding mode gives different results from truncation
* when the exact result is not an integer and is positive.
* <p>
* For examples, see {@link #ceilDiv(int, int)}.
*
* @param x the dividend
* @param y the divisor
* @return the smallest (closest to negative infinity)
* {@code long} value that is greater than or equal to the algebraic quotient.
* @throws ArithmeticException if the divisor {@code y} is zero
* @see #ceilMod(int, int)
* @see #ceil(double)
* @since 18
*/
public static long ceilDiv(long x, int y) {
return ceilDiv(x, (long)y);
}
/**
* Returns the smallest (closest to negative infinity)
* {@code long} value that is greater than or equal to the algebraic quotient.
* There is one special case: if the dividend is
* {@linkplain Long#MIN_VALUE Long.MIN_VALUE} and the divisor is {@code -1},
* then integer overflow occurs and
* the result is equal to {@code Long.MIN_VALUE}.
* <p>
* Normal integer division operates under the round to zero rounding mode
* (truncation). This operation instead acts under the round toward
* positive infinity (ceiling) rounding mode.
* The ceiling rounding mode gives different results from truncation
* when the exact result is not an integer and is positive.
* <p>
* For examples, see {@link #ceilDiv(int, int)}.
*
* @param x the dividend
* @param y the divisor
* @return the smallest (closest to negative infinity)
* {@code long} value that is greater than or equal to the algebraic quotient.
* @throws ArithmeticException if the divisor {@code y} is zero
* @see #ceilMod(int, int)
* @see #ceil(double)
* @since 18
*/
public static long ceilDiv(long x, long y) {
final long q = x / y;
// if the signs are the same and modulo not zero, round up
if ((x ^ y) >= 0 && (q * y != x)) {
return q + 1;
}
return q;
}
/**
* Returns the ceiling modulus of the {@code int} arguments.
* <p>
* The ceiling modulus is {@code r = x - (ceilDiv(x, y) * y)},
* has the opposite sign as the divisor {@code y} or is zero, and
* is in the range of {@code -abs(y) < r < +abs(y)}.
*
* <p>
* The relationship between {@code ceilDiv} and {@code ceilMod} is such that:
* <ul>
* <li>{@code ceilDiv(x, y) * y + ceilMod(x, y) == x}</li>
* </ul>
* <p>
* The difference in values between {@code ceilMod} and the {@code %} operator
* is due to the difference between {@code ceilDiv} and the {@code /}
* operator, as detailed in {@linkplain #ceilDiv(int, int)}.
* <p>
* Examples:
* <ul>
* <li>Regardless of the signs of the arguments, {@code ceilMod}(x, y)
* is zero exactly when {@code x % y} is zero as well.</li>
* <li>If neither {@code ceilMod}(x, y) nor {@code x % y} is zero,
* they differ exactly when the signs of the arguments are the same.<br>
* <ul>
* <li>{@code ceilMod(+4, +3) == -2}; &nbsp; and {@code (+4 % +3) == +1}</li>
* <li>{@code ceilMod(-4, -3) == +2}; &nbsp; and {@code (-4 % -3) == -1}</li>
* <li>{@code ceilMod(+4, -3) == +1}; &nbsp; and {@code (+4 % -3) == +1}</li>
* <li>{@code ceilMod(-4, +3) == -1}; &nbsp; and {@code (-4 % +3) == -1}</li>
* </ul>
* </li>
* </ul>
*
* @param x the dividend
* @param y the divisor
* @return the ceiling modulus {@code x - (ceilDiv(x, y) * y)}
* @throws ArithmeticException if the divisor {@code y} is zero
* @see #ceilDiv(int, int)
* @since 18
*/
public static int ceilMod(int x, int y) {
final int r = x % y;
// if the signs are the same and modulo not zero, adjust result
if ((x ^ y) >= 0 && r != 0) {
return r - y;
}
return r;
}
/**
* Returns the ceiling modulus of the {@code long} and {@code int} arguments.
* <p>
* The ceiling modulus is {@code r = x - (ceilDiv(x, y) * y)},
* has the opposite sign as the divisor {@code y} or is zero, and
* is in the range of {@code -abs(y) < r < +abs(y)}.
*
* <p>
* The relationship between {@code ceilDiv} and {@code ceilMod} is such that:
* <ul>
* <li>{@code ceilDiv(x, y) * y + ceilMod(x, y) == x}</li>
* </ul>
* <p>
* For examples, see {@link #ceilMod(int, int)}.
*
* @param x the dividend
* @param y the divisor
* @return the ceiling modulus {@code x - (ceilDiv(x, y) * y)}
* @throws ArithmeticException if the divisor {@code y} is zero
* @see #ceilDiv(long, int)
* @since 18
*/
public static int ceilMod(long x, int y) {
// Result cannot overflow the range of int.
return (int)ceilMod(x, (long)y);
}
/**
* Returns the ceiling modulus of the {@code long} arguments.
* <p>
* The ceiling modulus is {@code r = x - (ceilDiv(x, y) * y)},
* has the opposite sign as the divisor {@code y} or is zero, and
* is in the range of {@code -abs(y) < r < +abs(y)}.
*
* <p>
* The relationship between {@code ceilDiv} and {@code ceilMod} is such that:
* <ul>
* <li>{@code ceilDiv(x, y) * y + ceilMod(x, y) == x}</li>
* </ul>
* <p>
* For examples, see {@link #ceilMod(int, int)}.
*
* @param x the dividend
* @param y the divisor
* @return the ceiling modulus {@code x - (ceilDiv(x, y) * y)}
* @throws ArithmeticException if the divisor {@code y} is zero
* @see #ceilDiv(long, long)
* @since 18
*/
public static long ceilMod(long x, long y) {
final long r = x % y;
// if the signs are the same and modulo not zero, adjust result
if ((x ^ y) >= 0 && r != 0) {
return r - y;
}
return r;
}
/**
* Returns the absolute value of an {@code int} value.
* If the argument is not negative, the argument is returned.
* If the argument is negative, the negation of the argument is returned.
*
* <p>Note that if the argument is equal to the value of {@link
* Integer#MIN_VALUE}, the most negative representable {@code int}
* value, the result is that same value, which is negative. In
* contrast, the {@link Math#absExact(int)} method throws an
* {@code ArithmeticException} for this value.
*
* @param a the argument whose absolute value is to be determined
* @return the absolute value of the argument.
* @see Math#absExact(int)
*/
@IntrinsicCandidate
public static int abs(int a) {
return (a < 0) ? -a : a;
}
/**
* Returns the mathematical absolute value of an {@code int} value
* if it is exactly representable as an {@code int}, throwing
* {@code ArithmeticException} if the result overflows the
* positive {@code int} range.
*
* <p>Since the range of two's complement integers is asymmetric
* with one additional negative value (JLS {@jls 4.2.1}), the
* mathematical absolute value of {@link Integer#MIN_VALUE}
* overflows the positive {@code int} range, so an exception is
* thrown for that argument.
*
* @param a the argument whose absolute value is to be determined
* @return the absolute value of the argument, unless overflow occurs
* @throws ArithmeticException if the argument is {@link Integer#MIN_VALUE}
* @see Math#abs(int)
* @since 15
*/
public static int absExact(int a) {
if (a == Integer.MIN_VALUE)
throw new ArithmeticException(
"Overflow to represent absolute value of Integer.MIN_VALUE");
else
return abs(a);
}
/**
* Returns the absolute value of a {@code long} value.
* If the argument is not negative, the argument is returned.
* If the argument is negative, the negation of the argument is returned.
*
* <p>Note that if the argument is equal to the value of {@link
* Long#MIN_VALUE}, the most negative representable {@code long}
* value, the result is that same value, which is negative. In
* contrast, the {@link Math#absExact(long)} method throws an
* {@code ArithmeticException} for this value.
*
* @param a the argument whose absolute value is to be determined
* @return the absolute value of the argument.
* @see Math#absExact(long)
*/
@IntrinsicCandidate
public static long abs(long a) {
return (a < 0) ? -a : a;
}
/**
* Returns the mathematical absolute value of an {@code long} value
* if it is exactly representable as an {@code long}, throwing
* {@code ArithmeticException} if the result overflows the
* positive {@code long} range.
*
* <p>Since the range of two's complement integers is asymmetric
* with one additional negative value (JLS {@jls 4.2.1}), the
* mathematical absolute value of {@link Long#MIN_VALUE} overflows
* the positive {@code long} range, so an exception is thrown for
* that argument.
*
* @param a the argument whose absolute value is to be determined
* @return the absolute value of the argument, unless overflow occurs
* @throws ArithmeticException if the argument is {@link Long#MIN_VALUE}
* @see Math#abs(long)
* @since 15
*/
public static long absExact(long a) {
if (a == Long.MIN_VALUE)
throw new ArithmeticException(
"Overflow to represent absolute value of Long.MIN_VALUE");
else
return abs(a);
}
/**
* Returns the absolute value of a {@code float} value.
* If the argument is not negative, the argument is returned.
* If the argument is negative, the negation of the argument is returned.
* Special cases:
* <ul><li>If the argument is positive zero or negative zero, the
* result is positive zero.
* <li>If the argument is infinite, the result is positive infinity.
* <li>If the argument is NaN, the result is NaN.</ul>
*
* @apiNote As implied by the above, one valid implementation of
* this method is given by the expression below which computes a
* {@code float} with the same exponent and significand as the
* argument but with a guaranteed zero sign bit indicating a
* positive value:<br>
* {@code Float.intBitsToFloat(0x7fffffff & Float.floatToRawIntBits(a))}
*
* @param a the argument whose absolute value is to be determined
* @return the absolute value of the argument.
*/
@IntrinsicCandidate
public static float abs(float a) {
// Convert to bit field form, zero the sign bit, and convert back
return Float.intBitsToFloat(Float.floatToRawIntBits(a) & FloatConsts.MAG_BIT_MASK);
}
/**
* Returns the absolute value of a {@code double} value.
* If the argument is not negative, the argument is returned.
* If the argument is negative, the negation of the argument is returned.
* Special cases:
* <ul><li>If the argument is positive zero or negative zero, the result
* is positive zero.
* <li>If the argument is infinite, the result is positive infinity.
* <li>If the argument is NaN, the result is NaN.</ul>
*
* @apiNote As implied by the above, one valid implementation of
* this method is given by the expression below which computes a
* {@code double} with the same exponent and significand as the
* argument but with a guaranteed zero sign bit indicating a
* positive value:<br>
* {@code Double.longBitsToDouble((Double.doubleToRawLongBits(a)<<1)>>>1)}
*
* @param a the argument whose absolute value is to be determined
* @return the absolute value of the argument.
*/
@IntrinsicCandidate
public static double abs(double a) {
// Convert to bit field form, zero the sign bit, and convert back
return Double.longBitsToDouble(Double.doubleToRawLongBits(a) & DoubleConsts.MAG_BIT_MASK);
}
/**
* Returns the greater of two {@code int} values. That is, the
* result is the argument closer to the value of
* {@link Integer#MAX_VALUE}. If the arguments have the same value,
* the result is that same value.
*
* @param a an argument.
* @param b another argument.
* @return the larger of {@code a} and {@code b}.
*/
@IntrinsicCandidate
public static int max(int a, int b) {
return (a >= b) ? a : b;
}
/**
* Returns the greater of two {@code long} values. That is, the
* result is the argument closer to the value of
* {@link Long#MAX_VALUE}. If the arguments have the same value,
* the result is that same value.
*
* @param a an argument.
* @param b another argument.
* @return the larger of {@code a} and {@code b}.
*/
@IntrinsicCandidate
public static long max(long a, long b) {
return (a >= b) ? a : b;
}
// Use raw bit-wise conversions on guaranteed non-NaN arguments.
private static final long negativeZeroFloatBits = Float.floatToRawIntBits(-0.0f);
private static final long negativeZeroDoubleBits = Double.doubleToRawLongBits(-0.0d);
/**
* Returns the greater of two {@code float} values. That is,
* the result is the argument closer to positive infinity. If the
* arguments have the same value, the result is that same
* value. If either value is NaN, then the result is NaN. Unlike
* the numerical comparison operators, this method considers
* negative zero to be strictly smaller than positive zero. If one
* argument is positive zero and the other negative zero, the
* result is positive zero.
*
* @apiNote
* This method corresponds to the maximum operation defined in
* IEEE 754.
*
* @param a an argument.
* @param b another argument.
* @return the larger of {@code a} and {@code b}.
*/
@IntrinsicCandidate
public static float max(float a, float b) {
if (a != a)
return a; // a is NaN
if ((a == 0.0f) &&
(b == 0.0f) &&
(Float.floatToRawIntBits(a) == negativeZeroFloatBits)) {
// Raw conversion ok since NaN can't map to -0.0.
return b;
}
return (a >= b) ? a : b;
}
/**
* Returns the greater of two {@code double} values. That
* is, the result is the argument closer to positive infinity. If
* the arguments have the same value, the result is that same
* value. If either value is NaN, then the result is NaN. Unlike
* the numerical comparison operators, this method considers
* negative zero to be strictly smaller than positive zero. If one
* argument is positive zero and the other negative zero, the
* result is positive zero.
*
* @apiNote
* This method corresponds to the maximum operation defined in
* IEEE 754.
*
* @param a an argument.
* @param b another argument.
* @return the larger of {@code a} and {@code b}.
*/
@IntrinsicCandidate
public static double max(double a, double b) {
if (a != a)
return a; // a is NaN
if ((a == 0.0d) &&
(b == 0.0d) &&
(Double.doubleToRawLongBits(a) == negativeZeroDoubleBits)) {
// Raw conversion ok since NaN can't map to -0.0.
return b;
}
return (a >= b) ? a : b;
}
/**
* Returns the smaller of two {@code int} values. That is,
* the result the argument closer to the value of
* {@link Integer#MIN_VALUE}. If the arguments have the same
* value, the result is that same value.
*
* @param a an argument.
* @param b another argument.
* @return the smaller of {@code a} and {@code b}.
*/
@IntrinsicCandidate
public static int min(int a, int b) {
return (a <= b) ? a : b;
}
/**
* Returns the smaller of two {@code long} values. That is,
* the result is the argument closer to the value of
* {@link Long#MIN_VALUE}. If the arguments have the same
* value, the result is that same value.
*
* @param a an argument.
* @param b another argument.
* @return the smaller of {@code a} and {@code b}.
*/
@IntrinsicCandidate
public static long min(long a, long b) {
return (a <= b) ? a : b;
}
/**
* Returns the smaller of two {@code float} values. That is,
* the result is the value closer to negative infinity. If the
* arguments have the same value, the result is that same
* value. If either value is NaN, then the result is NaN. Unlike
* the numerical comparison operators, this method considers
* negative zero to be strictly smaller than positive zero. If
* one argument is positive zero and the other is negative zero,
* the result is negative zero.
*
* @apiNote
* This method corresponds to the minimum operation defined in
* IEEE 754.
*
* @param a an argument.
* @param b another argument.
* @return the smaller of {@code a} and {@code b}.
*/
@IntrinsicCandidate
public static float min(float a, float b) {
if (a != a)
return a; // a is NaN
if ((a == 0.0f) &&
(b == 0.0f) &&
(Float.floatToRawIntBits(b) == negativeZeroFloatBits)) {
// Raw conversion ok since NaN can't map to -0.0.
return b;
}
return (a <= b) ? a : b;
}
/**
* Returns the smaller of two {@code double} values. That
* is, the result is the value closer to negative infinity. If the
* arguments have the same value, the result is that same
* value. If either value is NaN, then the result is NaN. Unlike
* the numerical comparison operators, this method considers
* negative zero to be strictly smaller than positive zero. If one
* argument is positive zero and the other is negative zero, the
* result is negative zero.
*
* @apiNote
* This method corresponds to the minimum operation defined in
* IEEE 754.
*
* @param a an argument.
* @param b another argument.
* @return the smaller of {@code a} and {@code b}.
*/
@IntrinsicCandidate
public static double min(double a, double b) {
if (a != a)
return a; // a is NaN
if ((a == 0.0d) &&
(b == 0.0d) &&
(Double.doubleToRawLongBits(b) == negativeZeroDoubleBits)) {
// Raw conversion ok since NaN can't map to -0.0.
return b;
}
return (a <= b) ? a : b;
}
/**
* Clamps the value to fit between min and max. If the value is less
* than {@code min}, then {@code min} is returned. If the value is greater
* than {@code max}, then {@code max} is returned. Otherwise, the original
* value is returned.
* <p>
* While the original value of type long may not fit into the int type,
* the bounds have the int type, so the result always fits the int type.
* This allows to use method to safely cast long value to int with
* saturation.
*
* @param value value to clamp
* @param min minimal allowed value
* @param max maximal allowed value
* @return a clamped value that fits into {@code min..max} interval
* @throws IllegalArgumentException if {@code min > max}
*
* @since 21
*/
public static int clamp(long value, int min, int max) {
if (min > max) {
throw new IllegalArgumentException(min + " > " + max);
}
return (int) Math.min(max, Math.max(value, min));
}
/**
* Clamps the value to fit between min and max. If the value is less
* than {@code min}, then {@code min} is returned. If the value is greater
* than {@code max}, then {@code max} is returned. Otherwise, the original
* value is returned.
*
* @param value value to clamp
* @param min minimal allowed value
* @param max maximal allowed value
* @return a clamped value that fits into {@code min..max} interval
* @throws IllegalArgumentException if {@code min > max}
*
* @since 21
*/
public static long clamp(long value, long min, long max) {
if (min > max) {
throw new IllegalArgumentException(min + " > " + max);
}
return Math.min(max, Math.max(value, min));
}
/**
* Clamps the value to fit between min and max. If the value is less
* than {@code min}, then {@code min} is returned. If the value is greater
* than {@code max}, then {@code max} is returned. Otherwise, the original
* value is returned. If value is NaN, the result is also NaN.
* <p>
* Unlike the numerical comparison operators, this method considers
* negative zero to be strictly smaller than positive zero.
* E.g., {@code clamp(-0.0, 0.0, 1.0)} returns 0.0.
*
* @param value value to clamp
* @param min minimal allowed value
* @param max maximal allowed value
* @return a clamped value that fits into {@code min..max} interval
* @throws IllegalArgumentException if either of {@code min} and {@code max}
* arguments is NaN, or {@code min > max}, or {@code min} is +0.0, and
* {@code max} is -0.0.
*
* @since 21
*/
public static double clamp(double value, double min, double max) {
// This unusual condition allows keeping only one branch
// on common path when min < max and neither of them is NaN.
// If min == max, we should additionally check for +0.0/-0.0 case,
// so we're still visiting the if statement.
if (!(min < max)) { // min greater than, equal to, or unordered with respect to max; NaN values are unordered
if (Double.isNaN(min)) {
throw new IllegalArgumentException("min is NaN");
}
if (Double.isNaN(max)) {
throw new IllegalArgumentException("max is NaN");
}
if (Double.compare(min, max) > 0) {
throw new IllegalArgumentException(min + " > " + max);
}
// Fall-through if min and max are exactly equal (or min = -0.0 and max = +0.0)
// and none of them is NaN
}
return Math.min(max, Math.max(value, min));
}
/**
* Clamps the value to fit between min and max. If the value is less
* than {@code min}, then {@code min} is returned. If the value is greater
* than {@code max}, then {@code max} is returned. Otherwise, the original
* value is returned. If value is NaN, the result is also NaN.
* <p>
* Unlike the numerical comparison operators, this method considers
* negative zero to be strictly smaller than positive zero.
* E.g., {@code clamp(-0.0f, 0.0f, 1.0f)} returns 0.0f.
*
* @param value value to clamp
* @param min minimal allowed value
* @param max maximal allowed value
* @return a clamped value that fits into {@code min..max} interval
* @throws IllegalArgumentException if either of {@code min} and {@code max}
* arguments is NaN, or {@code min > max}, or {@code min} is +0.0f, and
* {@code max} is -0.0f.
*
* @since 21
*/
public static float clamp(float value, float min, float max) {
// This unusual condition allows keeping only one branch
// on common path when min < max and neither of them is NaN.
// If min == max, we should additionally check for +0.0/-0.0 case,
// so we're still visiting the if statement.
if (!(min < max)) { // min greater than, equal to, or unordered with respect to max; NaN values are unordered
if (Float.isNaN(min)) {
throw new IllegalArgumentException("min is NaN");
}
if (Float.isNaN(max)) {
throw new IllegalArgumentException("max is NaN");
}
if (Float.compare(min, max) > 0) {
throw new IllegalArgumentException(min + " > " + max);
}
// Fall-through if min and max are exactly equal (or min = -0.0 and max = +0.0)
// and none of them is NaN
}
return Math.min(max, Math.max(value, min));
}
/**
* Returns the fused multiply add of the three arguments; that is,
* returns the exact product of the first two arguments summed
* with the third argument and then rounded once to the nearest
* {@code double}.
*
* The rounding is done using the {@linkplain
* java.math.RoundingMode#HALF_EVEN round to nearest even
* rounding mode}.
*
* In contrast, if {@code a * b + c} is evaluated as a regular
* floating-point expression, two rounding errors are involved,
* the first for the multiply operation, the second for the
* addition operation.
*
* <p>Special cases:
* <ul>
* <li> If any argument is NaN, the result is NaN.
*
* <li> If one of the first two arguments is infinite and the
* other is zero, the result is NaN.
*
* <li> If the exact product of the first two arguments is infinite
* (in other words, at least one of the arguments is infinite and
* the other is neither zero nor NaN) and the third argument is an
* infinity of the opposite sign, the result is NaN.
*
* </ul>
*
* <p>Note that {@code fma(a, 1.0, c)} returns the same
* result as ({@code a + c}). However,
* {@code fma(a, b, +0.0)} does <em>not</em> always return the
* same result as ({@code a * b}) since
* {@code fma(-0.0, +0.0, +0.0)} is {@code +0.0} while
* ({@code -0.0 * +0.0}) is {@code -0.0}; {@code fma(a, b, -0.0)} is
* equivalent to ({@code a * b}) however.
*
* @apiNote This method corresponds to the fusedMultiplyAdd
* operation defined in IEEE 754.
*
* @param a a value
* @param b a value
* @param c a value
*
* @return (<i>a</i>&nbsp;&times;&nbsp;<i>b</i>&nbsp;+&nbsp;<i>c</i>)
* computed, as if with unlimited range and precision, and rounded
* once to the nearest {@code double} value
*
* @since 9
*/
@IntrinsicCandidate
public static double fma(double a, double b, double c) {
/*
* Infinity and NaN arithmetic is not quite the same with two
* roundings as opposed to just one so the simple expression
* "a * b + c" cannot always be used to compute the correct
* result. With two roundings, the product can overflow and
* if the addend is infinite, a spurious NaN can be produced
* if the infinity from the overflow and the infinite addend
* have opposite signs.
*/
// First, screen for and handle non-finite input values whose
// arithmetic is not supported by BigDecimal.
if (Double.isNaN(a) || Double.isNaN(b) || Double.isNaN(c)) {
return Double.NaN;
} else { // All inputs non-NaN
boolean infiniteA = Double.isInfinite(a);
boolean infiniteB = Double.isInfinite(b);
boolean infiniteC = Double.isInfinite(c);
double result;
if (infiniteA || infiniteB || infiniteC) {
if (infiniteA && b == 0.0 ||
infiniteB && a == 0.0 ) {
return Double.NaN;
}
double product = a * b;
if (Double.isInfinite(product) && !infiniteA && !infiniteB) {
// Intermediate overflow; might cause a
// spurious NaN if added to infinite c.
assert Double.isInfinite(c);
return c;
} else {
result = product + c;
assert !Double.isFinite(result);
return result;
}
} else { // All inputs finite
BigDecimal product = (new BigDecimal(a)).multiply(new BigDecimal(b));
if (c == 0.0) { // Positive or negative zero
// If the product is an exact zero, use a
// floating-point expression to compute the sign
// of the zero final result. The product is an
// exact zero if and only if at least one of a and
// b is zero.
if (a == 0.0 || b == 0.0) {
return a * b + c;
} else {
// The sign of a zero addend doesn't matter if
// the product is nonzero. The sign of a zero
// addend is not factored in the result if the
// exact product is nonzero but underflows to
// zero; see IEEE-754 2008 section 6.3 "The
// sign bit".
return product.doubleValue();
}
} else {
return product.add(new BigDecimal(c)).doubleValue();
}
}
}
}
/**
* Returns the fused multiply add of the three arguments; that is,
* returns the exact product of the first two arguments summed
* with the third argument and then rounded once to the nearest
* {@code float}.
*
* The rounding is done using the {@linkplain
* java.math.RoundingMode#HALF_EVEN round to nearest even
* rounding mode}.
*
* In contrast, if {@code a * b + c} is evaluated as a regular
* floating-point expression, two rounding errors are involved,
* the first for the multiply operation, the second for the
* addition operation.
*
* <p>Special cases:
* <ul>
* <li> If any argument is NaN, the result is NaN.
*
* <li> If one of the first two arguments is infinite and the
* other is zero, the result is NaN.
*
* <li> If the exact product of the first two arguments is infinite
* (in other words, at least one of the arguments is infinite and
* the other is neither zero nor NaN) and the third argument is an
* infinity of the opposite sign, the result is NaN.
*
* </ul>
*
* <p>Note that {@code fma(a, 1.0f, c)} returns the same
* result as ({@code a + c}). However,
* {@code fma(a, b, +0.0f)} does <em>not</em> always return the
* same result as ({@code a * b}) since
* {@code fma(-0.0f, +0.0f, +0.0f)} is {@code +0.0f} while
* ({@code -0.0f * +0.0f}) is {@code -0.0f}; {@code fma(a, b, -0.0f)} is
* equivalent to ({@code a * b}) however.
*
* @apiNote This method corresponds to the fusedMultiplyAdd
* operation defined in IEEE 754.
*
* @param a a value
* @param b a value
* @param c a value
*
* @return (<i>a</i>&nbsp;&times;&nbsp;<i>b</i>&nbsp;+&nbsp;<i>c</i>)
* computed, as if with unlimited range and precision, and rounded
* once to the nearest {@code float} value
*
* @since 9
*/
@IntrinsicCandidate
public static float fma(float a, float b, float c) {
if (Float.isFinite(a) && Float.isFinite(b) && Float.isFinite(c)) {
if (a == 0.0 || b == 0.0) {
return a * b + c; // Handled signed zero cases
} else {
return (new BigDecimal((double)a * (double)b) // Exact multiply
.add(new BigDecimal((double)c))) // Exact sum
.floatValue(); // One rounding
// to a float value
}
} else {
// At least one of a,b, and c is non-finite. The result
// will be non-finite as well and will be the same
// non-finite value under double as float arithmetic.
return (float)fma((double)a, (double)b, (double)c);
}
}
/**
* Returns the size of an ulp of the argument. An ulp, unit in
* the last place, of a {@code double} value is the positive
* distance between this floating-point value and the {@code
* double} value next larger in magnitude. Note that for non-NaN
* <i>x</i>, <code>ulp(-<i>x</i>) == ulp(<i>x</i>)</code>.
*
* <p>Special Cases:
* <ul>
* <li> If the argument is NaN, then the result is NaN.
* <li> If the argument is positive or negative infinity, then the
* result is positive infinity.
* <li> If the argument is positive or negative zero, then the result is
* {@code Double.MIN_VALUE}.
* <li> If the argument is &plusmn;{@code Double.MAX_VALUE}, then
* the result is equal to 2<sup>971</sup>.
* </ul>
*
* @param d the floating-point value whose ulp is to be returned
* @return the size of an ulp of the argument
* @author Joseph D. Darcy
* @since 1.5
*/
public static double ulp(double d) {
int exp = getExponent(d);
return switch(exp) {
case Double.MAX_EXPONENT + 1 -> Math.abs(d); // NaN or infinity
case Double.MIN_EXPONENT - 1 -> Double.MIN_VALUE; // zero or subnormal
default -> {
assert exp <= Double.MAX_EXPONENT && exp >= Double.MIN_EXPONENT;
// ulp(x) is usually 2^(SIGNIFICAND_WIDTH-1)*(2^ilogb(x))
exp = exp - (DoubleConsts.SIGNIFICAND_WIDTH - 1);
if (exp >= Double.MIN_EXPONENT) {
yield powerOfTwoD(exp);
} else {
// return a subnormal result; left shift integer
// representation of Double.MIN_VALUE appropriate
// number of positions
yield Double.longBitsToDouble(1L <<
(exp - (Double.MIN_EXPONENT - (DoubleConsts.SIGNIFICAND_WIDTH - 1))));
}
}
};
}
/**
* Returns the size of an ulp of the argument. An ulp, unit in
* the last place, of a {@code float} value is the positive
* distance between this floating-point value and the {@code
* float} value next larger in magnitude. Note that for non-NaN
* <i>x</i>, <code>ulp(-<i>x</i>) == ulp(<i>x</i>)</code>.
*
* <p>Special Cases:
* <ul>
* <li> If the argument is NaN, then the result is NaN.
* <li> If the argument is positive or negative infinity, then the
* result is positive infinity.
* <li> If the argument is positive or negative zero, then the result is
* {@code Float.MIN_VALUE}.
* <li> If the argument is &plusmn;{@code Float.MAX_VALUE}, then
* the result is equal to 2<sup>104</sup>.
* </ul>
*
* @param f the floating-point value whose ulp is to be returned
* @return the size of an ulp of the argument
* @author Joseph D. Darcy
* @since 1.5
*/
public static float ulp(float f) {
int exp = getExponent(f);
return switch(exp) {
case Float.MAX_EXPONENT + 1 -> Math.abs(f); // NaN or infinity
case Float.MIN_EXPONENT - 1 -> Float.MIN_VALUE; // zero or subnormal
default -> {
assert exp <= Float.MAX_EXPONENT && exp >= Float.MIN_EXPONENT;
// ulp(x) is usually 2^(SIGNIFICAND_WIDTH-1)*(2^ilogb(x))
exp = exp - (FloatConsts.SIGNIFICAND_WIDTH - 1);
if (exp >= Float.MIN_EXPONENT) {
yield powerOfTwoF(exp);
} else {
// return a subnormal result; left shift integer
// representation of FloatConsts.MIN_VALUE appropriate
// number of positions
yield Float.intBitsToFloat(1 <<
(exp - (Float.MIN_EXPONENT - (FloatConsts.SIGNIFICAND_WIDTH - 1))));
}
}
};
}
/**
* Returns the signum function of the argument; zero if the argument
* is zero, 1.0 if the argument is greater than zero, -1.0 if the
* argument is less than zero.
*
* <p>Special Cases:
* <ul>
* <li> If the argument is NaN, then the result is NaN.
* <li> If the argument is positive zero or negative zero, then the
* result is the same as the argument.
* </ul>
*
* @param d the floating-point value whose signum is to be returned
* @return the signum function of the argument
* @author Joseph D. Darcy
* @since 1.5
*/
@IntrinsicCandidate
public static double signum(double d) {
return (d == 0.0 || Double.isNaN(d))?d:copySign(1.0, d);
}
/**
* Returns the signum function of the argument; zero if the argument
* is zero, 1.0f if the argument is greater than zero, -1.0f if the
* argument is less than zero.
*
* <p>Special Cases:
* <ul>
* <li> If the argument is NaN, then the result is NaN.
* <li> If the argument is positive zero or negative zero, then the
* result is the same as the argument.
* </ul>
*
* @param f the floating-point value whose signum is to be returned
* @return the signum function of the argument
* @author Joseph D. Darcy
* @since 1.5
*/
@IntrinsicCandidate
public static float signum(float f) {
return (f == 0.0f || Float.isNaN(f))?f:copySign(1.0f, f);
}
/**
* Returns the hyperbolic sine of a {@code double} value.
* The hyperbolic sine of <i>x</i> is defined to be
* (<i>e<sup>x</sup>&nbsp;&minus;&nbsp;e<sup>&minus;x</sup></i>)/2
* where <i>e</i> is {@linkplain Math#E Euler's number}.
*
* <p>Special cases:
* <ul>
*
* <li>If the argument is NaN, then the result is NaN.
*
* <li>If the argument is infinite, then the result is an infinity
* with the same sign as the argument.
*
* <li>If the argument is zero, then the result is a zero with the
* same sign as the argument.
*
* </ul>
*
* <p>The computed result must be within 2.5 ulps of the exact result.
*
* @param x The number whose hyperbolic sine is to be returned.
* @return The hyperbolic sine of {@code x}.
* @since 1.5
*/
public static double sinh(double x) {
return StrictMath.sinh(x);
}
/**
* Returns the hyperbolic cosine of a {@code double} value.
* The hyperbolic cosine of <i>x</i> is defined to be
* (<i>e<sup>x</sup>&nbsp;+&nbsp;e<sup>&minus;x</sup></i>)/2
* where <i>e</i> is {@linkplain Math#E Euler's number}.
*
* <p>Special cases:
* <ul>
*
* <li>If the argument is NaN, then the result is NaN.
*
* <li>If the argument is infinite, then the result is positive
* infinity.
*
* <li>If the argument is zero, then the result is {@code 1.0}.
*
* </ul>
*
* <p>The computed result must be within 2.5 ulps of the exact result.
*
* @param x The number whose hyperbolic cosine is to be returned.
* @return The hyperbolic cosine of {@code x}.
* @since 1.5
*/
public static double cosh(double x) {
return StrictMath.cosh(x);
}
/**
* Returns the hyperbolic tangent of a {@code double} value.
* The hyperbolic tangent of <i>x</i> is defined to be
* (<i>e<sup>x</sup>&nbsp;&minus;&nbsp;e<sup>&minus;x</sup></i>)/(<i>e<sup>x</sup>&nbsp;+&nbsp;e<sup>&minus;x</sup></i>),
* in other words, {@linkplain Math#sinh
* sinh(<i>x</i>)}/{@linkplain Math#cosh cosh(<i>x</i>)}. Note
* that the absolute value of the exact tanh is always less than
* 1.
*
* <p>Special cases:
* <ul>
*
* <li>If the argument is NaN, then the result is NaN.
*
* <li>If the argument is zero, then the result is a zero with the
* same sign as the argument.
*
* <li>If the argument is positive infinity, then the result is
* {@code +1.0}.
*
* <li>If the argument is negative infinity, then the result is
* {@code -1.0}.
*
* </ul>
*
* <p>The computed result must be within 2.5 ulps of the exact result.
* The result of {@code tanh} for any finite input must have
* an absolute value less than or equal to 1. Note that once the
* exact result of tanh is within 1/2 of an ulp of the limit value
* of &plusmn;1, correctly signed &plusmn;{@code 1.0} should
* be returned.
*
* @param x The number whose hyperbolic tangent is to be returned.
* @return The hyperbolic tangent of {@code x}.
* @since 1.5
*/
@IntrinsicCandidate
public static double tanh(double x) {
return StrictMath.tanh(x);
}
/**
* Returns sqrt(<i>x</i><sup>2</sup>&nbsp;+<i>y</i><sup>2</sup>)
* without intermediate overflow or underflow.
*
* <p>Special cases:
* <ul>
*
* <li> If either argument is infinite, then the result
* is positive infinity.
*
* <li> If either argument is NaN and neither argument is infinite,
* then the result is NaN.
*
* <li> If both arguments are zero, the result is positive zero.
* </ul>
*
* <p>The computed result must be within 1.5 ulps of the exact
* result. If one parameter is held constant, the results must be
* semi-monotonic in the other parameter.
*
* @param x a value
* @param y a value
* @return sqrt(<i>x</i><sup>2</sup>&nbsp;+<i>y</i><sup>2</sup>)
* without intermediate overflow or underflow
* @since 1.5
*/
public static double hypot(double x, double y) {
return StrictMath.hypot(x, y);
}
/**
* Returns <i>e</i><sup>x</sup>&nbsp;&minus;1. Note that for values of
* <i>x</i> near 0, the exact sum of
* {@code expm1(x)}&nbsp;+&nbsp;1 is much closer to the true
* result of <i>e</i><sup>x</sup> than {@code exp(x)}.
*
* <p>Special cases:
* <ul>
* <li>If the argument is NaN, the result is NaN.
*
* <li>If the argument is positive infinity, then the result is
* positive infinity.
*
* <li>If the argument is negative infinity, then the result is
* -1.0.
*
* <li>If the argument is zero, then the result is a zero with the
* same sign as the argument.
*
* </ul>
*
* <p>The computed result must be within 1 ulp of the exact result.
* Results must be semi-monotonic. The result of
* {@code expm1} for any finite input must be greater than or
* equal to {@code -1.0}. Note that once the exact result of
* <i>e</i><sup>{@code x}</sup>&nbsp;-&nbsp;1 is within 1/2
* ulp of the limit value -1, {@code -1.0} should be
* returned.
*
* @param x the exponent to raise <i>e</i> to in the computation of
* <i>e</i><sup>{@code x}</sup>&nbsp;&minus;1.
* @return the value <i>e</i><sup>{@code x}</sup>&nbsp;-&nbsp;1.
* @since 1.5
*/
public static double expm1(double x) {
return StrictMath.expm1(x);
}
/**
* Returns the natural logarithm of the sum of the argument and 1.
* Note that for small values {@code x}, the result of
* {@code log1p(x)} is much closer to the true result of ln(1
* + {@code x}) than the floating-point evaluation of
* {@code log(1.0+x)}.
*
* <p>Special cases:
*
* <ul>
*
* <li>If the argument is NaN or less than -1, then the result is
* NaN.
*
* <li>If the argument is positive infinity, then the result is
* positive infinity.
*
* <li>If the argument is negative one, then the result is
* negative infinity.
*
* <li>If the argument is zero, then the result is a zero with the
* same sign as the argument.
*
* </ul>
*
* <p>The computed result must be within 1 ulp of the exact result.
* Results must be semi-monotonic.
*
* @param x a value
* @return the value ln({@code x}&nbsp;+&nbsp;1), the natural
* log of {@code x}&nbsp;+&nbsp;1
* @since 1.5
*/
public static double log1p(double x) {
return StrictMath.log1p(x);
}
/**
* Returns the first floating-point argument with the sign of the
* second floating-point argument. Note that unlike the {@link
* StrictMath#copySign(double, double) StrictMath.copySign}
* method, this method does not require NaN {@code sign}
* arguments to be treated as positive values; implementations are
* permitted to treat some NaN arguments as positive and other NaN
* arguments as negative to allow greater performance.
*
* @apiNote
* This method corresponds to the copySign operation defined in
* IEEE 754.
*
* @param magnitude the parameter providing the magnitude of the result
* @param sign the parameter providing the sign of the result
* @return a value with the magnitude of {@code magnitude}
* and the sign of {@code sign}.
* @since 1.6
*/
@IntrinsicCandidate
public static double copySign(double magnitude, double sign) {
return Double.longBitsToDouble((Double.doubleToRawLongBits(sign) &
(DoubleConsts.SIGN_BIT_MASK)) |
(Double.doubleToRawLongBits(magnitude) &
(DoubleConsts.EXP_BIT_MASK |
DoubleConsts.SIGNIF_BIT_MASK)));
}
/**
* Returns the first floating-point argument with the sign of the
* second floating-point argument. Note that unlike the {@link
* StrictMath#copySign(float, float) StrictMath.copySign}
* method, this method does not require NaN {@code sign}
* arguments to be treated as positive values; implementations are
* permitted to treat some NaN arguments as positive and other NaN
* arguments as negative to allow greater performance.
*
* @apiNote
* This method corresponds to the copySign operation defined in
* IEEE 754.
*
* @param magnitude the parameter providing the magnitude of the result
* @param sign the parameter providing the sign of the result
* @return a value with the magnitude of {@code magnitude}
* and the sign of {@code sign}.
* @since 1.6
*/
@IntrinsicCandidate
public static float copySign(float magnitude, float sign) {
return Float.intBitsToFloat((Float.floatToRawIntBits(sign) &
(FloatConsts.SIGN_BIT_MASK)) |
(Float.floatToRawIntBits(magnitude) &
(FloatConsts.EXP_BIT_MASK |
FloatConsts.SIGNIF_BIT_MASK)));
}
/**
* Returns the unbiased exponent used in the representation of a
* {@code float}. Special cases:
*
* <ul>
* <li>If the argument is NaN or infinite, then the result is
* {@link Float#MAX_EXPONENT} + 1.
* <li>If the argument is zero or subnormal, then the result is
* {@link Float#MIN_EXPONENT} - 1.
* </ul>
* @apiNote
* This method is analogous to the logB operation defined in IEEE
* 754, but returns a different value on subnormal arguments.
*
* @param f a {@code float} value
* @return the unbiased exponent of the argument
* @since 1.6
*/
public static int getExponent(float f) {
/*
* Bitwise convert f to integer, mask out exponent bits, shift
* to the right and then subtract out float's bias adjust to
* get true exponent value
*/
return ((Float.floatToRawIntBits(f) & FloatConsts.EXP_BIT_MASK) >>
(FloatConsts.SIGNIFICAND_WIDTH - 1)) - FloatConsts.EXP_BIAS;
}
/**
* Returns the unbiased exponent used in the representation of a
* {@code double}. Special cases:
*
* <ul>
* <li>If the argument is NaN or infinite, then the result is
* {@link Double#MAX_EXPONENT} + 1.
* <li>If the argument is zero or subnormal, then the result is
* {@link Double#MIN_EXPONENT} - 1.
* </ul>
* @apiNote
* This method is analogous to the logB operation defined in IEEE
* 754, but returns a different value on subnormal arguments.
*
* @param d a {@code double} value
* @return the unbiased exponent of the argument
* @since 1.6
*/
public static int getExponent(double d) {
/*
* Bitwise convert d to long, mask out exponent bits, shift
* to the right and then subtract out double's bias adjust to
* get true exponent value.
*/
return (int)(((Double.doubleToRawLongBits(d) & DoubleConsts.EXP_BIT_MASK) >>
(DoubleConsts.SIGNIFICAND_WIDTH - 1)) - DoubleConsts.EXP_BIAS);
}
/**
* Returns the floating-point number adjacent to the first
* argument in the direction of the second argument. If both
* arguments compare as equal the second argument is returned.
*
* <p>
* Special cases:
* <ul>
* <li> If either argument is a NaN, then NaN is returned.
*
* <li> If both arguments are signed zeros, {@code direction}
* is returned unchanged (as implied by the requirement of
* returning the second argument if the arguments compare as
* equal).
*
* <li> If {@code start} is
* &plusmn;{@link Double#MIN_VALUE} and {@code direction}
* has a value such that the result should have a smaller
* magnitude, then a zero with the same sign as {@code start}
* is returned.
*
* <li> If {@code start} is infinite and
* {@code direction} has a value such that the result should
* have a smaller magnitude, {@link Double#MAX_VALUE} with the
* same sign as {@code start} is returned.
*
* <li> If {@code start} is equal to &plusmn;
* {@link Double#MAX_VALUE} and {@code direction} has a
* value such that the result should have a larger magnitude, an
* infinity with same sign as {@code start} is returned.
* </ul>
*
* @param start starting floating-point value
* @param direction value indicating which of
* {@code start}'s neighbors or {@code start} should
* be returned
* @return The floating-point number adjacent to {@code start} in the
* direction of {@code direction}.
* @since 1.6
*/
public static double nextAfter(double start, double direction) {
/*
* The cases:
*
* nextAfter(+infinity, 0) == MAX_VALUE
* nextAfter(+infinity, +infinity) == +infinity
* nextAfter(-infinity, 0) == -MAX_VALUE
* nextAfter(-infinity, -infinity) == -infinity
*
* are naturally handled without any additional testing
*/
/*
* IEEE 754 floating-point numbers are lexicographically
* ordered if treated as signed-magnitude integers.
* Since Java's integers are two's complement,
* incrementing the two's complement representation of a
* logically negative floating-point value *decrements*
* the signed-magnitude representation. Therefore, when
* the integer representation of a floating-point value
* is negative, the adjustment to the representation is in
* the opposite direction from what would initially be expected.
*/
// Branch to descending case first as it is more costly than ascending
// case due to start != 0.0d conditional.
if (start > direction) { // descending
if (start != 0.0d) {
final long transducer = Double.doubleToRawLongBits(start);
return Double.longBitsToDouble(transducer + ((transducer > 0L) ? -1L : 1L));
} else { // start == 0.0d && direction < 0.0d
return -Double.MIN_VALUE;
}
} else if (start < direction) { // ascending
// Add +0.0 to get rid of a -0.0 (+0.0 + -0.0 => +0.0)
// then bitwise convert start to integer.
final long transducer = Double.doubleToRawLongBits(start + 0.0d);
return Double.longBitsToDouble(transducer + ((transducer >= 0L) ? 1L : -1L));
} else if (start == direction) {
return direction;
} else { // isNaN(start) || isNaN(direction)
return start + direction;
}
}
/**
* Returns the floating-point number adjacent to the first
* argument in the direction of the second argument. If both
* arguments compare as equal a value equivalent to the second argument
* is returned.
*
* <p>
* Special cases:
* <ul>
* <li> If either argument is a NaN, then NaN is returned.
*
* <li> If both arguments are signed zeros, a value equivalent
* to {@code direction} is returned.
*
* <li> If {@code start} is
* &plusmn;{@link Float#MIN_VALUE} and {@code direction}
* has a value such that the result should have a smaller
* magnitude, then a zero with the same sign as {@code start}
* is returned.
*
* <li> If {@code start} is infinite and
* {@code direction} has a value such that the result should
* have a smaller magnitude, {@link Float#MAX_VALUE} with the
* same sign as {@code start} is returned.
*
* <li> If {@code start} is equal to &plusmn;
* {@link Float#MAX_VALUE} and {@code direction} has a
* value such that the result should have a larger magnitude, an
* infinity with same sign as {@code start} is returned.
* </ul>
*
* @param start starting floating-point value
* @param direction value indicating which of
* {@code start}'s neighbors or {@code start} should
* be returned
* @return The floating-point number adjacent to {@code start} in the
* direction of {@code direction}.
* @since 1.6
*/
public static float nextAfter(float start, double direction) {
/*
* The cases:
*
* nextAfter(+infinity, 0) == MAX_VALUE
* nextAfter(+infinity, +infinity) == +infinity
* nextAfter(-infinity, 0) == -MAX_VALUE
* nextAfter(-infinity, -infinity) == -infinity
*
* are naturally handled without any additional testing
*/
/*
* IEEE 754 floating-point numbers are lexicographically
* ordered if treated as signed-magnitude integers.
* Since Java's integers are two's complement,
* incrementing the two's complement representation of a
* logically negative floating-point value *decrements*
* the signed-magnitude representation. Therefore, when
* the integer representation of a floating-point value
* is negative, the adjustment to the representation is in
* the opposite direction from what would initially be expected.
*/
// Branch to descending case first as it is more costly than ascending
// case due to start != 0.0f conditional.
if (start > direction) { // descending
if (start != 0.0f) {
final int transducer = Float.floatToRawIntBits(start);
return Float.intBitsToFloat(transducer + ((transducer > 0) ? -1 : 1));
} else { // start == 0.0f && direction < 0.0f
return -Float.MIN_VALUE;
}
} else if (start < direction) { // ascending
// Add +0.0 to get rid of a -0.0 (+0.0 + -0.0 => +0.0)
// then bitwise convert start to integer.
final int transducer = Float.floatToRawIntBits(start + 0.0f);
return Float.intBitsToFloat(transducer + ((transducer >= 0) ? 1 : -1));
} else if (start == direction) {
return (float)direction;
} else { // isNaN(start) || isNaN(direction)
return start + (float)direction;
}
}
/**
* Returns the floating-point value adjacent to {@code d} in
* the direction of positive infinity. This method is
* semantically equivalent to {@code nextAfter(d,
* Double.POSITIVE_INFINITY)}; however, a {@code nextUp}
* implementation may run faster than its equivalent
* {@code nextAfter} call.
*
* <p>Special Cases:
* <ul>
* <li> If the argument is NaN, the result is NaN.
*
* <li> If the argument is positive infinity, the result is
* positive infinity.
*
* <li> If the argument is zero, the result is
* {@link Double#MIN_VALUE}
*
* </ul>
*
* @apiNote This method corresponds to the nextUp
* operation defined in IEEE 754.
*
* @param d starting floating-point value
* @return The adjacent floating-point value closer to positive
* infinity.
* @since 1.6
*/
public static double nextUp(double d) {
// Use a single conditional and handle the likely cases first.
if (d < Double.POSITIVE_INFINITY) {
// Add +0.0 to get rid of a -0.0 (+0.0 + -0.0 => +0.0).
final long transducer = Double.doubleToRawLongBits(d + 0.0D);
return Double.longBitsToDouble(transducer + ((transducer >= 0L) ? 1L : -1L));
} else { // d is NaN or +Infinity
return d;
}
}
/**
* Returns the floating-point value adjacent to {@code f} in
* the direction of positive infinity. This method is
* semantically equivalent to {@code nextAfter(f,
* Float.POSITIVE_INFINITY)}; however, a {@code nextUp}
* implementation may run faster than its equivalent
* {@code nextAfter} call.
*
* <p>Special Cases:
* <ul>
* <li> If the argument is NaN, the result is NaN.
*
* <li> If the argument is positive infinity, the result is
* positive infinity.
*
* <li> If the argument is zero, the result is
* {@link Float#MIN_VALUE}
*
* </ul>
*
* @apiNote This method corresponds to the nextUp
* operation defined in IEEE 754.
*
* @param f starting floating-point value
* @return The adjacent floating-point value closer to positive
* infinity.
* @since 1.6
*/
public static float nextUp(float f) {
// Use a single conditional and handle the likely cases first.
if (f < Float.POSITIVE_INFINITY) {
// Add +0.0 to get rid of a -0.0 (+0.0 + -0.0 => +0.0).
final int transducer = Float.floatToRawIntBits(f + 0.0F);
return Float.intBitsToFloat(transducer + ((transducer >= 0) ? 1 : -1));
} else { // f is NaN or +Infinity
return f;
}
}
/**
* Returns the floating-point value adjacent to {@code d} in
* the direction of negative infinity. This method is
* semantically equivalent to {@code nextAfter(d,
* Double.NEGATIVE_INFINITY)}; however, a
* {@code nextDown} implementation may run faster than its
* equivalent {@code nextAfter} call.
*
* <p>Special Cases:
* <ul>
* <li> If the argument is NaN, the result is NaN.
*
* <li> If the argument is negative infinity, the result is
* negative infinity.
*
* <li> If the argument is zero, the result is
* {@code -Double.MIN_VALUE}
*
* </ul>
*
* @apiNote This method corresponds to the nextDown
* operation defined in IEEE 754.
*
* @param d starting floating-point value
* @return The adjacent floating-point value closer to negative
* infinity.
* @since 1.8
*/
public static double nextDown(double d) {
if (Double.isNaN(d) || d == Double.NEGATIVE_INFINITY)
return d;
else {
if (d == 0.0)
return -Double.MIN_VALUE;
else
return Double.longBitsToDouble(Double.doubleToRawLongBits(d) +
((d > 0.0d)?-1L:+1L));
}
}
/**
* Returns the floating-point value adjacent to {@code f} in
* the direction of negative infinity. This method is
* semantically equivalent to {@code nextAfter(f,
* Float.NEGATIVE_INFINITY)}; however, a
* {@code nextDown} implementation may run faster than its
* equivalent {@code nextAfter} call.
*
* <p>Special Cases:
* <ul>
* <li> If the argument is NaN, the result is NaN.
*
* <li> If the argument is negative infinity, the result is
* negative infinity.
*
* <li> If the argument is zero, the result is
* {@code -Float.MIN_VALUE}
*
* </ul>
*
* @apiNote This method corresponds to the nextDown
* operation defined in IEEE 754.
*
* @param f starting floating-point value
* @return The adjacent floating-point value closer to negative
* infinity.
* @since 1.8
*/
public static float nextDown(float f) {
if (Float.isNaN(f) || f == Float.NEGATIVE_INFINITY)
return f;
else {
if (f == 0.0f)
return -Float.MIN_VALUE;
else
return Float.intBitsToFloat(Float.floatToRawIntBits(f) +
((f > 0.0f)?-1:+1));
}
}
private static final double F_UP = 0x1p1023; // normal, exact, 2^DoubleConsts.EXP_BIAS
private static final double F_DOWN = 0x1p-1023; // subnormal, exact, 2^-DoubleConsts.EXP_BIAS
/**
* Returns {@code d} &times; 2<sup>{@code scaleFactor}</sup>
* rounded as if performed by a single correctly rounded
* floating-point multiply. If the exponent of the result is
* between {@link Double#MIN_EXPONENT} and {@link
* Double#MAX_EXPONENT}, the answer is calculated exactly. If the
* exponent of the result would be larger than {@code
* Double.MAX_EXPONENT}, an infinity is returned. Note that if
* the result is subnormal, precision may be lost; that is, when
* {@code scalb(x, n)} is subnormal, {@code scalb(scalb(x, n),
* -n)} may not equal <i>x</i>. When the result is non-NaN, the
* result has the same sign as {@code d}.
*
* <p>Special cases:
* <ul>
* <li> If the first argument is NaN, NaN is returned.
* <li> If the first argument is infinite, then an infinity of the
* same sign is returned.
* <li> If the first argument is zero, then a zero of the same
* sign is returned.
* </ul>
*
* @apiNote This method corresponds to the scaleB operation
* defined in IEEE 754.
*
* @param d number to be scaled by a power of two.
* @param scaleFactor power of 2 used to scale {@code d}
* @return {@code d} &times; 2<sup>{@code scaleFactor}</sup>
* @since 1.6
*/
public static double scalb(double d, int scaleFactor) {
if (scaleFactor > -DoubleConsts.EXP_BIAS) {
if (scaleFactor <= DoubleConsts.EXP_BIAS) {
return d * primPowerOfTwoD(scaleFactor);
}
if (scaleFactor <= 2 * DoubleConsts.EXP_BIAS) {
return d * primPowerOfTwoD(scaleFactor - DoubleConsts.EXP_BIAS) * F_UP;
}
if (scaleFactor < 2 * DoubleConsts.EXP_BIAS + PRECISION - 1) {
return d * primPowerOfTwoD(scaleFactor - 2 * DoubleConsts.EXP_BIAS) * F_UP * F_UP;
}
return d * F_UP * F_UP * F_UP;
}
if (scaleFactor > -2 * DoubleConsts.EXP_BIAS) {
return d * primPowerOfTwoD(scaleFactor + DoubleConsts.EXP_BIAS) * F_DOWN;
}
if (scaleFactor > -2 * DoubleConsts.EXP_BIAS - PRECISION) {
return d * primPowerOfTwoD(scaleFactor + 2 * DoubleConsts.EXP_BIAS) * F_DOWN * F_DOWN;
}
return d * MIN_VALUE * MIN_VALUE;
}
/**
* Returns {@code f} &times; 2<sup>{@code scaleFactor}</sup>
* rounded as if performed by a single correctly rounded
* floating-point multiply. If the exponent of the result is
* between {@link Float#MIN_EXPONENT} and {@link
* Float#MAX_EXPONENT}, the answer is calculated exactly. If the
* exponent of the result would be larger than {@code
* Float.MAX_EXPONENT}, an infinity is returned. Note that if the
* result is subnormal, precision may be lost; that is, when
* {@code scalb(x, n)} is subnormal, {@code scalb(scalb(x, n),
* -n)} may not equal <i>x</i>. When the result is non-NaN, the
* result has the same sign as {@code f}.
*
* <p>Special cases:
* <ul>
* <li> If the first argument is NaN, NaN is returned.
* <li> If the first argument is infinite, then an infinity of the
* same sign is returned.
* <li> If the first argument is zero, then a zero of the same
* sign is returned.
* </ul>
*
* @apiNote This method corresponds to the scaleB operation
* defined in IEEE 754.
*
* @param f number to be scaled by a power of two.
* @param scaleFactor power of 2 used to scale {@code f}
* @return {@code f} &times; 2<sup>{@code scaleFactor}</sup>
* @since 1.6
*/
public static float scalb(float f, int scaleFactor) {
// magnitude of a power of two so large that scaling a finite
// nonzero value by it would be guaranteed to over or
// underflow; due to rounding, scaling down takes an
// additional power of two which is reflected here
final int MAX_SCALE = Float.MAX_EXPONENT + -Float.MIN_EXPONENT +
FloatConsts.SIGNIFICAND_WIDTH + 1;
// Make sure scaling factor is in a reasonable range
scaleFactor = Math.max(Math.min(scaleFactor, MAX_SCALE), -MAX_SCALE);
/*
* Since + MAX_SCALE for float fits well within the double
* exponent range and + float -> double conversion is exact
* the multiplication below will be exact. Therefore, the
* rounding that occurs when the double product is cast to
* float will be the correctly rounded float result.
*/
return (float)((double)f*powerOfTwoD(scaleFactor));
}
/**
* Returns a floating-point power of two in the normal range.
*/
static double powerOfTwoD(int n) {
assert(n >= Double.MIN_EXPONENT && n <= Double.MAX_EXPONENT);
return primPowerOfTwoD(n);
}
/**
* Returns a floating-point power of two in the normal range.
* No checks are performed on the argument.
*/
private static double primPowerOfTwoD(int n) {
return longBitsToDouble((long) (n + DoubleConsts.EXP_BIAS) << PRECISION - 1);
}
/**
* Returns a floating-point power of two in the normal range.
*/
static float powerOfTwoF(int n) {
assert(n >= Float.MIN_EXPONENT && n <= Float.MAX_EXPONENT);
return Float.intBitsToFloat(((n + FloatConsts.EXP_BIAS) <<
(FloatConsts.SIGNIFICAND_WIDTH-1))
& FloatConsts.EXP_BIT_MASK);
}
/**
* Returns the product of the unsigned arguments,
* throwing an exception if the result overflows an unsigned {@code int}.
*
* @param x the first unsigned value
* @param y the second unsigned value
* @return the result
* @throws ArithmeticException if the result overflows an unsigned int
* @since 25
*/
public static int unsignedMultiplyExact(int x, int y) {
long r = (x & 0xFFFF_FFFFL) * (y & 0xFFFF_FFFFL);
if (r >>> 32 != 0) {
throw new ArithmeticException("unsigned integer overflow");
}
return (int)r;
}
/**
* Returns the product of the unsigned arguments,
* throwing an exception if the result overflows an unsigned {@code long}.
*
* @param x the first unsigned value
* @param y the second unsigned value
* @return the result
* @throws ArithmeticException if the result overflows an unsigned long
* @since 25
*/
public static long unsignedMultiplyExact(long x, int y) {
return unsignedMultiplyExact(x, y & 0xFFFF_FFFFL);
}
/**
* Returns the product of the unsigned arguments,
* throwing an exception if the result overflows an unsigned {@code long}.
*
* @param x the first unsigned value
* @param y the second unsigned value
* @return the result
* @throws ArithmeticException if the result overflows an unsigned long
* @since 25
*/
public static long unsignedMultiplyExact(long x, long y) {
long l = x * y;
long h = unsignedMultiplyHigh(x, y);
if (h == 0) {
return l;
}
throw new ArithmeticException("unsigned long overflow");
}
/**
* Returns {@code x} raised to the power of {@code n},
* throwing an exception if the result overflows an {@code int}.
* When {@code n} is 0, the returned value is 1.
*
* @param x the base.
* @param n the exponent.
* @return {@code x} raised to the power of {@code n}.
* @throws ArithmeticException when {@code n} is negative,
* or when the result overflows an int.
* @since 25
*/
public static int powExact(int x, int n) {
/* See the comment in unsignedPowExact(long,int) for the details. */
if (n < 0) {
throw new ArithmeticException("negative exponent");
}
if (n == 0) {
return 1;
}
if (x == 0 || x == 1) {
return x;
}
if (x == -1) {
return (n & 0b1) == 0 ? 1 : -1;
}
int p = 1;
while (n > 1) {
if ((n & 0b1) != 0) {
p *= x;
}
x = multiplyExact(x, x);
n >>>= 1;
}
return multiplyExact(p, x);
}
/**
* Returns unsigned {@code x} raised to the power of {@code n},
* throwing an exception if the result overflows an unsigned {@code int}.
* When {@code n} is 0, the returned value is 1.
*
* @param x the unsigned base.
* @param n the exponent.
* @return {@code x} raised to the power of {@code n}.
* @throws ArithmeticException when {@code n} is negative,
* or when the result overflows an unsigned int.
* @since 25
*/
public static int unsignedPowExact(int x, int n) {
/* See the comment in unsignedPowExact(long,int) for the details. */
if (n < 0) {
throw new ArithmeticException("negative exponent");
}
if (n == 0) {
return 1;
}
if (x == 0 || x == 1) {
return x;
}
int p = 1;
while (n > 1) {
if ((n & 0b1) != 0) {
p *= x;
}
x = unsignedMultiplyExact(x, x);
n >>>= 1;
}
return unsignedMultiplyExact(p, x);
}
/**
* Returns {@code x} raised to the power of {@code n},
* throwing an exception if the result overflows a {@code long}.
* When {@code n} is 0, the returned value is 1.
*
* @param x the base.
* @param n the exponent.
* @return {@code x} raised to the power of {@code n}.
* @throws ArithmeticException when {@code n} is negative,
* or when the result overflows a long.
* @since 25
*/
public static long powExact(long x, int n) {
/* See the comment in unsignedPowExact(long,int) for the details. */
if (n < 0) {
throw new ArithmeticException("negative exponent");
}
if (n == 0) {
return 1;
}
if (x == 0 || x == 1) {
return x;
}
if (x == -1) {
return (n & 0b1) != 0 ? -1 : 1;
}
long p = 1;
while (n > 1) {
if ((n & 0b1) != 0) {
p *= x;
}
x = multiplyExact(x, x);
n >>>= 1;
}
return multiplyExact(p, x);
}
/**
* Returns unsigned {@code x} raised to the power of {@code n},
* throwing an exception if the result overflows an unsigned {@code long}.
* When {@code n} is 0, the returned value is 1.
*
* @param x the unsigned base.
* @param n the exponent.
* @return {@code x} raised to the power of {@code n}.
* @throws ArithmeticException when {@code n} is negative,
* or when the result overflows an unsigned long.
* @since 25
*/
public static long unsignedPowExact(long x, int n) {
if (n < 0) {
throw new ArithmeticException("negative exponent");
}
if (n == 0) {
return 1;
}
/*
* To keep the code as simple as possible, there are intentionally
* no fast paths, except for |x| <= 1.
* The reason is that the number of loop iterations below can be kept
* very small when |x| > 1, but not necessarily when |x| <= 1.
*/
if (x == 0 || x == 1) {
return x;
}
/*
* Let x0 and n0 > 0 be the entry values of x and n, resp.
* The useful loop invariants are:
* p * x^n = x0^n0
* |p| < |x|
*
* Since |x0| >= 2 here, and since |x0|^(2^6) >= 2^Long.SIZE, the squaring
* of x in the loop overflows at latest during the 6th iteration,
* so by then the method throws.
* Thus, the loop executes at most 5 successful iterations, and fails
* not later than at the 6th.
*
* But n is right-shifted at each iteration.
* If the method returns, there are thus floor(log2(n0)) iterations.
*/
long p = 1;
while (n > 1) {
if ((n & 0b1) != 0) {
/*
* The invariant |p| < |x| holds, so we have |p*x| < |x*x|.
* That is, if p*x overflows, so does x*x below, which is
* always executed.
* In other words, a plain * can be used here, since we are
* piggybacking on the squaring of x to throw.
*/
p *= x;
}
x = unsignedMultiplyExact(x, x);
n >>>= 1;
}
return unsignedMultiplyExact(p, x);
}
}